A simulacrum cannot research spells
By the text of the spell, the simulacrum "lacks the ability to learn or become more powerful" (PHB p.276). It cannot learn new spells, and it certainly can't create new spells.
A simulacrum might be able to create magic items
This mostly depends on whether your DM allows NPCs to craft magic items following the same rules as PCs. By the text of the spell, the simulacrum "can take actions... as a normal creature," "uses all the statistics of the creature it duplicates," and it obeys your commands. So if a normal creature with your present capabilities can create a magic item, you can command it to do so.
As stated, the simulacrum cannot learn, so if your DM does allow NPCs to create items this way, it will only be able to create something you already have the ability to create on your own.
A homunculus probably can't take advantage of any kind of downtime activity
The homunculus description indicates that it acts as an extension of its creator, "sharing thoughts, senses, and language" (MM p.188). It's likely that any activity the homunculus takes on requires at least some of your focus. This is supported by the Shared Mind feature:
A homunculus knows everything its creator knows, including all the languages the creator can speak and read. Likewise, everything the construct senses is known to its master, even over great distances, provided both are on the same plane.
Note that the text indicates that the shared knowledge relationship goes one way. What the homunculus learns is not automatically transferred to the master; the master can experience its senses, but to actually benefit from a downtime activity, it is required that a certain amount of time and energy be spent focusing on that activity.
It is certainly possible to interpret this differently (a broad reading of "a spellcaster engaged in secret experimentation" would support your view, for example,) but the wording of the homunculus' description seems carefully written to make the master the active party when it comes to accumulating knowledge.