[RPG] When does a hidden opponent that is attacking become visible

Immediately after whatever caused them to lose the condition allowing them to Hide. That is sometimes "immediately", and sometimes after an action. Lets look at some examples:

Hide

Say you're using Hide to be hidden, because you have Cover or Concealment. It's daytime, or another source of full illumination. Here's the relevant part of Hide (emphasis mine):

You need cover or concealment in order to attempt a Hide check. Total cover or total concealment usually (but not always; see Special, below) obviates the need for a Hide check, since nothing can see you anyway.

If people are observing you, even casually, you can’t hide. You can run around a corner or behind cover so that you’re out of sight and then hide, but the others then know at least where you went.

If you walk out of cover/concealment and into the open where you can be seen, you're seen. Period. This happens immediately, as per Spot:

Action

Varies. Every time you have a chance to spot something in a reactive manner you can make a Spot check without using an action.

Vision in D&D is omnidirectional and immediate. A spot check in daylight has DC 0, so it's basically impossible to fail. You can't use Hide because you don't have cover or concealment.

What does that mean for Sneak Attack? You can't use it. You can't use it by walking out, you can't use it in a charge, or any other way. Unless you can attack from cover/concealment, you can't do this with Hide.

Note that you can Snipe, from range:

Sniping

If you’ve already successfully hidden at least 10 feet from your target, you can make one ranged attack, then immediately hide again. You take a -20 penalty on your Hide check to conceal yourself after the shot.

The reason why this works is that you can attack from behind cover at range, so the condition to attack while hidden (and then hide again) exists. They're "aware" of you, but they are denied their dexterity bonus to AC because they can't see you to defend against you (as you're hidden). In this case, you can Sneak Attack.

Hide in Plain Sight

Depending on how you get it, Hide in Plain Sight has some conditions. If you meet those, you can use Hide even if you don't have cover/concealment and often times if people are watching you. This makes Hide work far more effectively - You can hide, move up, and attack. Your first attack will deny them dex to AC (and thus enable Sneak Attack) if your hide check succeeds against their spot.

After the first attack, you'd have to Hide again (at a -20 penalty for attacking) to remain hidden. If you don't, any subsequent attack that round is not considered hidden. Thus, they get their dex bonus to AC and you can not Sneak Attack.

Note that using Hide after attacking is in most cases a move action, so your DM will probably not allow you to full attack while making Hide rolls after every single attack to remain hidden.

Invisibility

You can Sneak Attack with Invisibility. That said, Invisiblity ends after the first attack you make. That means if you use a full attack, only the first attack you make gets the benefit of Invisibility. The others are made normally, with the target having his dex bonus to AC and are not Sneak Attacks. Here's the rule on that:

Perhaps the most common form of total concealment is the invisibility spell. A regular invisibility effect is broken when you attack. If you begin your turn under such an effect and you're making multiple attacks, you'll be invisible only for the first attack and your opponent will be denied Dexterity bonus to Armor Class (if any) against that first attack.

So once again, the "stealth status" change takes effect instantly as conditions change in combat.

Greater Invisibility does not break when you attack, and so in that case you could keep Sneak Attacking.

Now to cover your specific cases that I haven't already covered:

If the stealthed character moves before attempting a sneak attack, does the foe gain the opportunity to detect them during the movement and before the attack triggers?

Yes. That's covered by Hide and Hide in Plain Sight, above. Any time the conditions change in regards to seeing the Hiding person, they're entitled to a check.

Do they have no such opportunity if no movement is done first?

If you're already hidden (eg, you have cover and your Hide beat their Spot), if you don't break cover and they don't move to break your cover, nothing changes. You're still hidden, and can Snipe. You'll have to make a new check to stay hidden after you Snipe, as described above.

Now, they might get a move silently check if you move, even if they already failed the Spot. Listen can give them an idea of where you are, but is not enough to change the conditions for Sneak Attack (being able to hear you to the left isn't enough to change things, it just tells them what direction to search in on their turn).

If its their turn, they can make another Spot check as a move action. This applies if they failed to see you and your conditions didn't change, so they don't get a reactive check.

Trying to spot something you failed to see previously is a move action.

Is this different if the movement and attack are a single action, such as a Charge?

No, it changes nothing. In fact Charge takes its own -20 penalty to Hide.

Remember that a full attack action is a "single action" as well, but things change during it. (This is different from Pathfinder and AFAIK 4e.)

Is there any case where i can Sneak Attack for multiple attacks?

Sure. If they're Flat-Footed at the start of combat, they're denied their dex until their turn comes around. If you can attack them before that, all your attacks are Sneak Attacks.

If you flank them, you will still flank them on subsequent attacks (barring something unusual going on).

As mentioned, Greater Invisibility doesn't break when you attack, so you can keep using it.

The key takeaway here is that conditions change in response to what you're doing. It doesn't wait to take effect. If Invisibility drops after the first of your four attacks during your full round action, it drops. The other three do not get its benefit.

There are 2 definitions of "round" in 5e. There is the definition of "round" that you quote, which is from initiative position intMax to initiative position 0 (or negative if you manage that some how).

Then there is the definition used here for readied actions. This is the same definition used in "once per round" effects such as certain powers. This definition begins at the beginning of your turn and ends at the beginning of your next turn.

Because (unlike in 4e), readying an action and activating it, doesn't move your initiative order position, and because it matches the readied action refresh timer (which happens to use the same 1/round definition of round), readied actions can be used at any point in either the current round or the next round prior to your turn.

Here's how reactions refresh:

When you take a reaction, you can’t take another one until the start of your next turn. If the reaction interrupts another creature’s turn, that creature can continue its turn right after the reaction. (Player's Basic p70)

This is a large part of the support that leads me (and many others) to believe that readied actions can roll up to your next turn.