According to the Basic PDF (page 74)
A creature's current hit points (usually just called hit points) can be any number from the creature's hit point maximum down to 0.
So no, negative hit points do not exist in 5e. The archived development-poll blog post "A Close Call with Negative Hit Points" explains the history of negative hit points and part of the reasoning for the change in 5e.
- Can a 5e simulacrum gain temporary hit points YES
- Can a 5e simulacrum regenerate an arcane ward YES
Simulacrum PHB p276
It appears to be the same as the original, but it has half the creature’s hit point maximum and is formed without any equipment. Otherwise, the illusion uses all the statistics of the creature it duplicates.
The simulacrum lacks the ability to learn or become more powerful, so it never increases its level or other abilities, nor can it regain expended spell slots.
If the simulacrum is damaged, you can repair it in an alchemical laboratory, using rare herbs and minerals worth 100 gp per hit point it regains. The simulacrum lasts until it drops to 0 hit points, at which point it reverts to snow and melts instantly.
The spell description about learning or becoming more powerful refers to implicit power i.e. level, training, hit points, spell slots. It does not refer to being the recipient of the effects from the use of spells or abilities that do not restore lost resources. Bless, Guidance, Enhance Ability, raging, Gauntlets of Ogre Power etc. all work on it. A healing potion does not. A Pearl of Power does not, and, to look at a different Wizard Tradition, Expert Divination (PHB p116) does not restore spell slots (RAW, though the actual description of the ability might lead to a house rule that the spell slot is not regained but actually represents it not having been used in the first place).
Temporary hit points PHB p198
Some spells and special abilities confer temporary hit points to a creature. Temporary hit points aren’t actual hit points; they are a buffer against damage, a pool of hit points that protect you from injury.
Temporary hit points "aren’t actual hit points" so gaining them does not represent recovering hit points in any way, they are an effect of a spell or ability so the simulacrum can gain the benefit, just as they can wear Gauntlets of Ogre power to give them 19 strength.
Arcane Ward PHB p115
you can weave magic around yourself for protection. When you cast an abjuration spell of 1st level or higher, you can simultaneously use a strand of the spell’s magic to create a magical ward on yourself that lasts until you finish a long rest. The ward has hit points equal to twice your wizard level + your Intelligence modifier. Whenever you take damage, the ward takes the damage instead.
The protection gained from this ability is a ward that has it's strength measured in hit points. The simulacrum does not gain or regain hit points with this ability, it gains a ward, an ability's effect, where "Whenever you take damage, the ward takes the damage instead" with it's effect measured in hit points, and it is the the ward that regains strength when the simulacrum casts certain spells.
As an aside, the ward's strength is not temporary hit points so the Simulacrum, or any other creature with this ability, could have those at the same time too.
Best Answer
Temporary Hit Points do not make you Conscious.
Rules Compendium p. 258 on Temporary Hit Points indicate that:
Rules Compendium p. 260 on Dying and Death indicate that:
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).