1. In Unity's character booklet, the Auxiliary Sheet says the following:
Unity can create mechanical minions, which follow the normal rules for minions. They act at the beginning of the turn, and only
have one kind of action they can take. The minions Unity makes in battle are rushed, and only stay operational for the current scene.
Afterward, they short out, fizzle, or melt into nothing, and cannot be harvested for parts.
Plus, according to the podcast, Christopher and Adam clarify that the bots Unity makes are only temporary and fall apart relatively soon after their creation. They are temporary creations from her mind and only have enough energy to be around for so long before they fall apart. In this case, it appears to be long enough for one fight. If your GM rules as written, no, they do not last beyond the few minutes that the fight happens, and that is in accordance with the Christopher-and-Adam canon.
2. How would Unity know what kind of bots she needs for the fight before the fight happens? Unity seems to me to be more the type to assess the situation and use the best tool for the job. If a fight isn't going down, then she wouldn't have any bots around "just in case." Besides, they would fall apart anyway given their short lifespan, so why would she waste any of that time just having them around? Far more efficient to maximize their usage and pull them together during the battle!
3. Philosophically, I would say that depends on the situation you are in. The card game and the RPG are cut from similar cloth, but use different mechanics. The card game represents one big fight. The RPG could have more than that in one issue, such that from the story perspective and in keeping with the Christopher-and-Adam canon Unity wouldn't be able to have an army of bots following her around the whole time. They break down, fall apart, and aren't meant to be around for much longer than minutes. Hours aren't possible. Functionally, you can't. You have to make a choice, bots or something else, and that is what an RPG is all about. Players make choices with the understanding that they have to give up something else when the choice is made. If you make a bot the first round, you have to define what one basic action that bot can do, which means you are giving up the four other basic actions for that bot. It also means that it would take five turns to get one bot of each basic action into the game, unless you take a risky action to make more than one bot a turn. If you don't want to neglect her other abilities, then do something other than just make bots. Bots are useful, but Unity can also shoot lightning at stuff with her superpowers. Doing so requires some cool creative description to assemble the dice pool, which makes playing Unity an exercise in imagination. Ultimately Unity is a support tech character at her core, whipping up what the team needs to help them succeed.
4. Simple. Make a Boost Bot. Start of next turn, have Boost Bot give itself a Bot Upgrade Bonus. Then make and upgrade a New Bot by discarding the Boost Bot's Bot Upgrade Bonus. If you wanted to raise the size of your bots, then you need a bunch of Boost Bots around, one to give itself the Bot Upgrade Bonus and the others to give you a bunch of Create Bot Bonuses to raise your bot making rolls. Similar to the card game, to activate the better options you first need to generate resources and slowly build over turns. This is another choice, though. Doing this means you aren't doing anything immediately contributing to the group, but that is exactly how Unity works in the card game too.
Now, all that being said, I agree with Timjobinroach. Talk to your GM. Ultimately they have final say on how Unity works. My feeling is that having a bunch of bots always around cheapens the fun and excitement of the game for everyone. Bunch of spider bots attacking? Unity's army takes care of it. Dinosaurs roaming around? Unity's army takes care of it. Stuck in a tomb full of traps? Unity's army takes care of it. Unity has the potential to unbalance the game not just for Unity's player, but for the entire table, and the answers your GM gives you can't be given lightly. There are a bunch of factors they need to consider, and while it might be fun for you to have a bunch of bots running around all the time, it might take some of the fun away from the other players who might have their own abilities they want to use, and force the GM to do more work than should be needed to accomodate and balance the challenges of the issue for the added bot army factor. In the card game, the balance is maintained by playing one card and using one power. You start out slow and hope to get to the point where you can do a whole lot of stuff on your turn. Here, you're asking your GM to allow you to start out with your end result already in play at the start of the fight, and I would encourage you talk to your GM about these questions, while keeping the full game experience for everyone in mind as you explore answers.