The UA on sidekicks has a lot of potential but I feel like it wasn't thought all the way through. Here are some obvious questions I have as a DM who wants to implement this.

1. Can a familiar/animal companion/paladin mount become a sidekick? If so, the beast master/chain warlock just became POWERFUL.

2. How does this affect encounter building? With the beast master, I treat the ranger/animal companion as one PC because the ranger is effectively sacrificing personal power for the animal (and the animal never hits PC levels of power). But this sidekick is about 75% of a full PC. Normally, I add allied NPCs into the party strength and I split the XP with that NPC. However, the sidekick levels with the PC/party. It seems to imply that it ignores XP like a beast master's pet.

3. What level do you start a sidekicked NPC at? If the level 12 paladin gets a squire, are they a level 1 warrior or level 12? Seems a bit unfair to limit them to level 1 but maybe a bit OP to have the squire suddenly be level 12.

Personally, I think the sidekick is a bit too powerful to have more than 1 per party. Unless the DM wants to greatly up his encounters (and bog down combat), they should impose this limit and maybe make the sidekick party controlled.

I think if I were to do multiple sidekicks, then I would give a sidekick level every other level. It makes them squisher but honestly, sidekicks should be a bit flimsy.

EDIT: I also would like rules for ordering an animal sidekick around. Right now, I'm using the beast master rules of "Use an action to give a command" but I don't know if the Beast Master should be better in that regard

My personal take on this:

1. No, I think the intent is for creatures that are not obtained via other means than "becoming friends". Also, the Sidekick rules state that the creature at base should be CR 1 or lower. Because an animal companion already gets bonuses that would probably require a recalculation of CR if it were a "wild" version of said creature, that would rule that out. Familiars might, but note that they can't attack. This means that the usefulness of a familiar sidekick is limited, but not impossible. Again, I don't think familiars should be sidekicks, but I don't see the harm in it.

2. Effectively, I think that the sidekicks count as a PC, since they're powerful enough to affect combat encounters in favor of the players.

3. I'm guessing they start at 1. After all, they're sidekicks and it would not really make sense in-world; oh, you're an Elf Commoner that we've befriended? Great, join our group and instantly gain access to lvl 8 spells since we're level 16. Also, the UA says that it gains "a sidekick class" and that it "levels when you do". RAI I think this means that they gain the first level of a class when they become sidekicks and only level up as you do.

Sidekicks are mostly there for pets and/or simple characters to join the group when the story requires them. It's funny, but in my last session my players "befriended" a Blue Guard Drake and basically turned it into a Sidekick even though those rules were not out yet. But it's most helpful for small parties; if a group of 3 characters want to have more epic adventures, they might want to have a couple of "extra characters" in order to be able to best more powerful adversaries. They could, using the standard rules, simply make more characters using full player rules. However, those rules tend to bog things down in terms of combat speed; if you're required to come up with tactics for two full characters, there are a lot of variables to take into account. These Sidekick rules simplify that.

Your variant of levelling sidekicks up every other level is a great idea if you want the player characters as the "true heroes", like Batman and the sidekick as a... well... sidekick... Like Robin. However, if you're more inclined to use them as I mentioned above (say Batman and Nightwing) you might want to level them up as the characters do. 

I'm guessing that the intent is that Sidekicks are intelligent or "well trained" enough that they if have their own turn (and since the Warrior gets advantage to Initiative at lvl 7 I'm pretty sure this is the case) then you don't need to order them around. Since talking is basically a non-action, you could also rule that it would not take an action to "command" or advise a Sidekick to a particular action. For less intelligent Sidekicks, I think your ruling is pretty fair, given that the Sidekick probably would follow a command until given other orders.

Hope that helps :)

1. I would say no.

The sidekick can be any type of creature with a stat block in the Monster Manual or another D&D book

To me that screams unmodified creatures, which excludes familiars, ranger companions and paladin mounts whose stat blocks are modified by class features.

2. I would treat them as a full PC - their powers might be somewhat limited, but they still can take a full turn, and the action pool is more important than having one less spell slots per day. Not to mention, the sidekicks are controlled by the player(s), which means they will (or could) use their abilities to synergy with the PCs, way more than PCs do with each other.

3. It would be 12th level. As per the description,

So even if the sidekick has not been with the paladin up to this point, they did their own training and are level 12 by now.