a, b, g, and now n are the various revisions of the IEEE 802.11 spec - basically, each new version is an upgrade over the previous one. All the letters in between there were lost in the shuffle - they didn't differentiate themselves enough to make ti worth while to put in consumer products.
That being said, N has been out for a while now - it was ratified in January '06, if memory serves. Prior to that, most major computer manufacturers and wireless router companies had what were called "draft n" cards available - basically, they were faster than G, but not guaranteed to be compatible with official N devices after the revision was ratified. Most routers and wireless cards have no problem connecting to b/g networks, and anything you've bought in the past year or so should have an n card in it.
n provides many benefits to end consumers (and FIRST robots) - longer range and increased data bandwidth being the two big ones. For FIRST, this means a single router has a much better chance of avoiding interference from robot frames and materials than a g router would. Further, providing teams with a g router now would be providing them with older technology, something that they would have to update in the next year or two as the availability of g routers became scarce (If you can even find them today).
So, hopefully that provides some insight into why FIRST probably decided on the router they did.
Now to answer your question...
It *should* be fine to set your router to utilize b/g/n instead of only n. Best case, you won't notice a difference. In the middle there might be some interference introduced. And worst case, it simply won't work and you'll have to set it back to n only. And in the least likely scenario, you'll set it to b/g/n and the world will explode. I doubt that will happen though
As for the gaming adapter (the part that sits on the robot) - make sure that is set up like FIRST instructs. You definitely don't want that acting anyway other than the way it should work at the competition, just to save the possible headache of your robot not working.