If your DM allows for servants to be counted as friends, then yes, they are eligible.
There's an additional requirement that isn't explicitly mentioned in your question. Together, the three requirements are:
- The sidekick can be any type of creature with a stat block in the Monster Manual or another D&D book.
- Its challenge rating must be 1 or lower.
- The two of you must be friends.
Starting with #1, does Find Familiar and Animate Dead create a creature with a stat block within a D&D book?
- Yes, both spells specify that they "create" a creature, and the creature options listed have stat blocks.
Moving on to #2, are the aforementioned creatures a challenge rating of 1 or less?
- Yes, every listed creature from both spells meet that requirement (including the added familiars from Warlock's Pact of the Chain).
Finally at #3, are the aforementioned creatures your friend?
- Nothing in either spell says they are your friend, or even that they aren't hostile towards you. They are simply described as servants who obey your commands. Whether or not you and your familiar/undead servant are able to build a friendship is entirely up to you and the DM. It's worth keeping in mind however that Zombies, Skeletons, Imps, and Quasits are evil aligned (everything else is unaligned), so depending on your character, building a "friendship" may not be possible.
Supposing your DM does allow sidekicks in general, and familiars/undead servants specifically as friends, then all prerequisites will have been met. In this case, the answer would be yes, they would be eligible to become a sidekick.
Regarding your second question, the UA rule-set doesn't address the situation whatsoever, which means ultimately it'll be up to your DM to decide. However, both spells have a duration of instantaneous, and Animate Dead's time limit only has to do with your control over the creature rather than it's existence, so assuming they continue to qualify as your friend that shouldn't be an issue. Familiars also don't die in the traditional sense, and can be resummoned, so it would be unlikely for them to lose progress over it, so long as it's the same familiar your resummon.