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.}

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, 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.

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.