4 sets of levels (that i have seen so far) have been White- 0-5 Green- 0-10 Blue- 0-15 Purple- 0-20
So in order to get the booster/implant slots you must recruit a higher tier character. But you should rush for a full GREEN team then use blue or green level 2 or 3 implants and get through at least half of the story.
Your characters get promoted by earning experience int the game.
No, they only get promoted to a certain level. Beyond that you currently need to purchase a new character. Like the poster above you said, there are levels.
The devs made it known they are planning to bring in "Boot Camps" that will allow you to boost your current characters so they can continue to level.
Right your character levels up with experience to there max level then you need to buy the next set from boot camp located in. Then you can continue leveling your characters to there new max level.