>What point in the process do the hair roller cages go on?
Usually never. If you want to use them, their purpose is to keep an early queen from killing the rest or if you're going to be out of town when you should be putting them in the nucs. You'll have much better luck if you get them in the mating nucs on day 14 from when the queen was confined (in normal weather or 13 in hot weather) and don't bother with the cages.
>I guess the starter hive can be any hive, just capture the queen, plop her into the Jenter box thing for a day then put her back with the population again, from what I understand.
That is not the starter hive. That's the "donor" or "mother" hive. You pick a hive where you like the qualities that the queen has (gentle, productive, healthy etc.) and confine the queen from that hive. The starter is usually either a queenless hive or a "swarm box" which is a lot of bees shaken into a well ventilated nuc with a frame of honey a frame of pollen maybe another of each, a wet sponge in the bottom or some water sprayed on the frames of honey. After two hours in the "swarm box" you graft and put in the queen cells.
>Also, as far as the finisher hive goes, what is it's role?
Once they have started the queen cells, you could move them to another hive and start another batch of queen cells with that starter. Finishers are sometimes queenright.
> Is this a smaller, queenless hive that will be taking care of queen cells until they emerge?
Usually it takes care of them until they are two days from emerging. Then the cells are put into mating nucs to emerge.
> How many frames of bees does this consist of?
How many queens do you want? The main thing, as far as quality of queens, is the density of bees and availability of food.
> Will a nuc be good if using 1 of those modified frames that hold the plugs?
Only if it's overflowing with bees. And the less bees the more the finisher is likely to tear down some or all of the cells.
> I guess when things are all said and done, if you use a nuc for the finisher hive, you can just queen it when finished making queens and you have a new colony.
I usually do, yes.
>The question that I have is it hard to get the queen to lay in the plastic cups?
Put the box in the hive several days before you plan to confine the queen. Put it in the middle of the brood nest. After you release the queen, leave it in the middle of the brood nest, preferably with open brood next to them so nurse bees will be available.