- 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.
You Are You; and So Is He. If a simulacrum you have created casts wish, both you and your simulacrum suffer the stress associated with casting the spell—including the risk of being forever unable to cast wish again. The inability to cast wish extends to any simulacrum you create in the future.
The above is taken from the D&D Adventurers League FAQ March 24, 2017, currently available here.
However, that stated, these rulings are for the balance and survival of Adventurers League, an organized play setting, and not explicitly the traditional table setting. Without explicit rulings like this on things the books don't cover, a nationally organized play association would not be able to run reasonable events with a common experience across all venues. Having a whole lot of "This DM let me do it at this event" isn't healthy to that ecosystem.
It is well understood by many players and DMs alike that rules as written for Simulacrum + Wish can be exploited for some serious imbalance in player agency and power: you can do anything you want without suffering the usual consequences. While I can't speak to the true intent of Adventurers League, this is likely why Adventurers League has made this decision despite there being no such restriction in the Player's Handbook to that spell combination.
By RAW it works because the Simulacrum is a separate creature, it just happens to have your resources and abilities.
By AL rules it doesn't work, because the Simulacrum is treated as you for the purposes of Wish.
Your DM has Rule 0, not even wizards can stop that!
Wizards doesn't have final say at your table. They give you the system, the basic foundations of the mechanical bits and pieces of your storytelling experience. Your DM can decide to use this ruling as an example of best practices and put it into practice in their game. Or they can decide that your game doesn't need the measures against Simulacrum + Wish synergy and allow players which earn those abilities/spells to be able to use them in that way.
Best Answer
I make the cut where conventional learning stops and extraordinary abilities start.
Any ruling is up to the DM, and this DM thinks it would be absurd if "learning" would extend to the inability of perception and recognition of colours, places where the simulacrum might walk where it has not seen, or the inability to follow commands because no one has commanded the original in exactly the same manner.
In basic English, "learning" means the active process of getting an understanding of something by studying it or by experience over time. So this DM thinks the best approach to defining the limitations is by looking at what the inability to learn signifies. The spell description implies:
The common factor is the acquisition of such abilities that are extraordinary and would create permanent resources in the form of these extraordinary abilities.
A simulacrum that doesn't remember commands and cannot follow them would be absurd. In the same manner, it would be absurd if it couldn't walk to places or obtain basic information about its surroundings. So this DM ruling is such that the limitations of learning concern the acquisition or replenishing of such abilities that are extraordinary and would create permanent resources in the form of abilities, feats, spell slots, traits or replenishing them once the simulacrum expends them.
Based on the above ruling, the simulacrum can remember names and what happens to them. I have also ruled that they can make better decisions due to applying information that they conventionally learn, such as running a shop, haggling with customers, remembering their names and the procedures of shop upkeep.
I make the cut where conventional learning stops and extraordinary abilities start. I can also understand other DMs who would rule that some "conventional learning" might be out of scope. This ruling is enjoyable for my players - the simulacrum is a resource that my players often deploy to enrich the storytelling. Limiting the simulacrum's ability to "conventionally learn," would limit the application of the spell. Such that it would remove almost every way that my players use the simulacrum. Removing the basic functions of "learning" would render the spell defunct.