Reading the description in context gives:
You shape an illusory duplicate of one beast or
humanoid that is within range for the entire casting
time of the spell. The duplicate is a creature, partially
real and formed from ice or snow, and it can take
actions and otherwise be affected as a normal creature. 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.
What the simulacrum is is an illusory beast or humanoid that is partially real - illusions don't heal naturally or magically.
Reading the monster manual everything is a creature - constructs, undead, beasts, huminoids etc. Being a creature does not mean it can heal naturally or magically - some creatures can and some can't.
The simulacrum has the statistics of the creature and half its hit points - it does not have any of the racial or class features except for spell slots that cannot be regained so healing surges and short rests don't come into it.
If the simulacrum is damaged, you can repair it ...
You can repair it - you can't heal it.
- 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
No, the alchemical process is the only way it can regain hit points
Repair not heal
First note that this does not say heal it says repair. Though treated as a normal creature it is still a being made of ice and snow. It is that reason that it likely has a special way to repair it.
Allowing other forms of healing means that the alchemical process would be useless
This statement must preclude other forms of regaining hit points otherwise it is meaningless. Even though it does not say so explicitly, the above method is clearly meant to be the only way to regain hit points for a simulacrum.
It does not say that potions or healing magic or rest doesn't work for example, but if they were possible then the expensive complicated process for healing would be meaningless and there would be no point in including it. Sleep especially makes this meaningless because it is free and available to every creature/class and allows for healing completely. Why would anybody ever pay 100gp per hit point if they could sleep or use any of the myriad of other methods for regaining hit points available? This logic also applies to traits or features that let you heal.
RAI agrees
Jeremy Crawford supports this logic as well:
Duplicated creature must be humanoid or beast
Also note that the creature you are duplicating must be a beast or humanoid and I could not find one of either in the Monster Manual that has regeneration. There may be ones from other sources, but it is something to be aware of.