Looking at RAW, no and maybe
To answer the first question:
I’m working under the assumption you’re not asking about summoned skulks – because summoned skulks don’t leave corpses in our plane, but rather they’re returned to the Shadowfell upon death (as stated in Mordekainen’s Tome of Foes, page 227):
A summoned skulk can’t return to the Shadowfell until it dies
The text for a skulk says:
They are so devoid of identity that they have become permanently invisible.
If the lack of identity was the key factor here, any summoned skulk would become visible automatically because they’d get a portion of the summoner’s identity. So, aside from the 3 ways to circumvent it, I’d rule the skulk’s corpse is still invisible.
Now onto your second question: animate dead says:
Choose a pile of bones or a corpse of a Medium or Small humanoid within range.
The first problem, if we go with the above that the skulk’s corpse is invisible, is how would you choose an invisible corpse? You’d have to use one of the 3 ways to circumvent the invisibility to even cast the spell on it.
Secondly, based on RAW and this answer the raised creature takes on the statistics of a skeleton or a zombie, so it would lose the invisibility to gain the abilities of a zombie.
If the DM wishes to retain the invisibility it would have to be a different template than a basic zombie or skeleton, and increases the challenge rating of such a creature. One suggestion would be to replace other zombie or skeleton abilities, or weaken the raised create further, if the goal is to retain the invisibility.