It's relatively well established that passive perception acts as a floor to an active perception check.

Does the same ruling apply to other skill/ability checks¹ (ie not explicitly perception)?

Are there any rules text that support this?

_{1: For the purposes of this question I'm assuming all skill checks can be made passively for the purposes of this question. If that assumption is wrong then that should also be addressed in an answer.}

No, not all passive ability checks are floors for active ability checks

Passive perception is a special case because it is generally assumed that you are always perceiving the world around you - so it makes sense that you always get the benefit of passive perception. For most skills, though, their use is not a background constant - if you try to jump a gap, you make one jump, and that might succeed or fail. You're not constantly trying to jump the gap such that it makes sense to take an average of all your jumping to see if you succeed or fail.

A good indicator that this is the way things work is that if passive ability checks were a floor for active checks, the Rogue class feature Reliable Talent would be almost useless:

By 11th level, you have refined your chosen skills until they approach perfection. Whenever you make an ability check that lets you add your proficiency bonus, you can treat a d20 roll of 9 or lower as a 10.

That is functionally almost the same as having your passive score function as a floor for your active ability checks. Given this is a special feature that the Rogue gains explicitly, it wouldn't make sense for that to also be the default rule for how checks work.

Only when applicable!

First, let's keep in mind passive and active DCs can vary. From LMoP:

Spotting a secret door from a distance of no more than 10 feet without actively searching for it requires a passive Wisdom (Perception) score of 15 or higher, whereas a character who takes the time to search the wall can find the secret door with a successful DC 10 Wisdom (Perception) check.

So, passive scores acting as a floor to a roll happens only when the DCs match (otherwise, no need to have different DCs). For the remainder of this answer, let's assume the DCs for active checks and passive scores match.

Now, when can we use passive scores? The rules are clear.

A passive check [...] can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster.

In the case of Perception, it is an ability that's always being done repeatedly by adventurers, and is described by the Hiding rules. As other answers have mentioned, about spotting hidden enemies, passive Perception is the floor:

When you hide, there’s a chance someone will notice you even if they aren’t searching. The GM compares your Stealth check with that creature’s passive Wisdom (Perception) score.

But can be superseeded by a better roll with an active search:

That check’s total is contested by the Wisdom (Perception) check of any creature that actively searches for signs of your presence.

That being said, I think it makes sense for some other scores to work passively. For example, Insight is my prime choice, as I assume PCs will notice by default if they're being lied to. When new monsters appear, Intelligence passive scores also tell PCs lore and names of enemies. If PCs then ask for active actions, then they may supersede the passive score.

However, actions like picking a lock, they're a short set of actions that perfectly match an active Sleight of Hand check. If he can just try and pick it all day, then I consider the passive score, but when they reach the house for the first time and try to break in, it's an active check, and not superseded by any passive scores.

Passive score first (if applicable). Then active check.

In the end, it depends on DM and PC. Use passive scores when it is something the PC is assumed to do by default, repeatedly, with no consequence (remain alert, ready to fight, recall information, maybe even seduce NPCs to affect their mood), and these may be superseded by active checks if PCs ask for them. Whenever it is not a repeatable task, and instead an actual event with consequences for failure (maybe as simple as wasted time), use active checks, since passive scores are not applicable.

Don't just assume all skills have a minimum. It would cause an 11th level Rogue skill to be useless.

No, passive ability checks should not (normally) be used as floors for ability checks

This might seem like a weird tangent initially, so bear with me...

At level 11 Rogue's gain the following ability:

Reliable Talent

By 11th level, you have refined your chosen skills until they approach perfection. Whenever you make an ability check that lets you add your proficiency bonus, you can treat a d20 roll of 9 or lower as a 10.

Passive ability checks are normally calculated via the formula:

10 + all modifiers that normally apply to the check

This is the same formula that Reliable Talent allows Level 11 Rogues to apply to any ability checks that they make which they are proficient in.

If passive ability checks were always the floor to normal ability checks then Reliable Talent would be completely useless. In applying only to ability checks you're proficient in, it's actually strictly worse than always using passive scores as a floor for any active check.

Passive ability scores as floors for active checks can therefore not be the games assumed default.

There are obviously exceptions:

Drawing from the basic rules, tasks done repeatedly (and presumingly time consumingly) can use a passive score:

A passive check [...] can represent the average result for a task done repeatedly, such as searching for secret doors over and over again,

Or can be used as a tool by the DM:

to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster.

But in general, for time pressured, single attempts at ability checks, a high passive score should not protect you from a fumble. Allowing it to do so breaks Reliable Talent.

No, passive scores are not floors to active skill checks.

No, specifically to your phrasing, "All active ability checks".

Active abilities, like Athletics and Acrobatics, always carry a chance of failure beyond "nothing happens." A DM will ask for rolls on these checks because their actions cannot be repeated and failure matters.

Passive abilities, like Perception, Insight, and (potentially) History, carry a "non-notice" result. Additionally, these checks are frequently made by the DM when determining what environmental qualities to describe. This is why passive scores are needed. They streamline the game by preventing players from rolling every time a new room is entered or an NPC encountered.

You state:

It's relatively well established that passive perception acts as a floor to an active perception check.

This is not the case.

Jeremy's guidance in the link does not suggest that PP is a floor to the skill's entire function. Anything a passive check would notice is not "lost" by a poor active check, as per Jeremy's suggestion, but the two may function differently.

In Jeremy's interview, he is discussing a check with similar DCs or results. This is not always the case.

As demonstrated in LMoP, for example, active and passive skill checks can function differently:

Spotting a secret door from a distance of no more than 10 feet without actively searching for it requires a passive Wisdom (Perception) score of 15 or higher, whereas a character who takes the time to search the wall can find the secret door with a successful DC 10 Wisdom (Perception) check.

To your question:

Are there any rules text that supports this?

The best guidance we know of on this comes from Jeremy's tweets/interviews, which at best offer this as an option to the DM.

Passive Skills

Passive skills have different functions than active skills,

Passive Perception is always "on", but does not necessarily produce the same results as an active perception check.

An active use of a skill is a time-consuming, focused application of the skill that should produce different results. Any active roll should always carry a chance of failure.

To follow the PHB (175) example:

A passive check is a special kind of ability check that doesn't involve any die rolls. Such a check can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice...

When the players are using a skill in an active manner, they are using it in different ways than the passive score. For example, players enter a room with a secret door and want to use perception.

The DM should provide information about the room, which might include features gated by passive perception scores. They players may decide:

• To find the door right now
  

• I want to search for the door and will take time

In the latter example, the DM is allowing the players an extended time to attempt an action that carries no catastrophic failure that can be repeated, or is providing information about the room. In DND 3E, this mechanic was called "taking 10." This is a special use of the passive score -- it allows the players to take the average.

In the former example, the players are attempting a single action with an individual result.

Yes, only if ...

If a DM decides (a) to use a passive check and (b) it's always active, it can function as a skill check minimum. Entirely up to the DM. Source: Crawford's tweet

There are two requirements which must be fulfilled before a passive skill can function as a skill check minimum. And again, JC reiterated that it is entirely up to the DM if they allow passive skill score to function as skill check minimum.

The reasoning is already stated by many other answers, either in this question and other question: you should compare passive score with passive DC (if it's different from the active DC). If passive score is equal or higher than the DC, you should not roll for active check and considered successful on the check.

Remember that this is only true if the passive check is always on. As a DM, I would only consider passive perception, insight, and investigation as "always on". No passive athletic or acrobatic, and I find passive stealth is not always on. Passive score is not the floor of skill check, but it effectively function as minimum because how a skill check is resolved: passive first, if not pass then try active.

Yes - effectively

I am not sure if there are specific rules to support this, but passive scores are always on, and are used before rolling.

In the case of Perception (an easy example) you either notice something passively, or you don't. You would never make an active check for something that you have already spotted.

For things you don't immediately spot then you can still make a check. That check can be lower than the passive perception, but overall since it is only rolled after passive perception fails it means the real value used is still the passive value.

The most common mistake that I see in DM's is ignoring passive values, you should only roll for something when there is a chance of failure, and if you can easily passively succeed there is no chance of failure - and there should be no roll.

Note: If each skill has a passive use or not is a different and very good question - I would be interested in seeing an answer on that matter.