[RPG] Can a zero-HP, unconscious and stable character be woken up prematurely

Temporary Hit Points do not make you Conscious.

Rules Compendium p. 258 on Temporary Hit Points indicate that:

Not Real Hit Points: Temporary hit points aren’t healing, but rather a layer of “insulation” that attacks have to get through before they start dealing real damage to a target. Don’t add temporary hit points to a creature’s current hit points (if an adventurer has 0 hit points or fewer, he or she still has 0 or fewer after receiving temporary hit points). Keep track of them as a separate pool of hit points.

Rules Compendium p. 260 on Dying and Death indicate that:

Dying: When an adventurer’s hit points drop to 0 or fewer, he or she falls unconscious and is dying.

Thus, when a character is at 0 hit points or fewer (including the -1 in your case) they are dying (a status keyword) and when they have this status, they are also unconscious unless otherwise specified. There are certain powers or features that explicitly state that a character is not "unconscious while dying" but the rules for temporary hit points lack this qualification, therefore they do not undo the unconscious while dying effect.

The character will have -1 hp and 5 thp. The first five points of incoming damage will still go to the temporary hit points before the rest goes to hit points. This means if the character is stabilized they will no longer make death saving throws even if they take up to 5 damage (so thp are useful on dying characters). But the character still has less than zero hp, is still dying, is still unconscious, and must still make death saving throws on his or her turn until stabilized (thp do not stabilize a dying character).

The entry for Death Saving Throw says:

Whenever you start your turn with 0 hit points....

So this just means that you make the throw at the beginning of your turn. However, nowhere in the description does it say that it requires any kind of action to make the save.

A saving throw is an instant response to a harmful effect and is almost never done by choice.

There is no general rule saying that doing any kind of saving throw (of which a death saving throw is one) takes any kind of action at all. Indeed the quote above would imply that it is not one. This especially makes sense since many saving throws are made not on your turn and thus would be unlikely to use your action or movement or reaction. Additionally, if saving throws did take any kind of action to resolve, it would be quite a huge power boost for those abilities that cause them.

That being the case, once you regain consciousness, you should have all of your move and action available to you to use as you see fit.

A good point from @MivaScott:

Most unconscious people are also prone. So while they are now conscious, I would still apply the "prone" condition for that turn (half movement length for crawling, get attacked with advantage, etc).