XJT FA here.
There is no, to my knowledge, FAA rule about what gets gate checked. I certainly could be wrong (and if I am, please correct me with the FAR #) but general rule of thumb is:
Per TSA: Two carry-on bags.
Like every airline: One goes under your seat, the other goes in the overhead
Per FAA: Absolutely no bags in your lap. No laptops in seatback pockets (please business travelers, I would think you knew this one by now.. ). This is so there is nothing in your lap/at your knees that can fly up and smack you in the face. I've seen it happen. I had to fight hard to keep from laughing (because I had told the same woman twice to stow her purse). It's the same reason your tray table needs to be up.
Per company policy: Items that don't fit and/or exceed the two carryon rule are gatechecked.
Here's how I roll:
If you enter the aircraft and I see a blue/green tag on your bag I will ask "Are you usually able to fit that bag on this size aircraft?". If they've never flown and we aren't running late, I'll give them the opportunity to try. If not, I'll explain that the bins are small and unfortunately that means gate-checking items.
Items I know wont fit, leave it over there to gate-check it.
Just please, please for the sake of your flight attendant (and an on-time departure) when placing those wheeled briefcase/computer bags into the bins, do them handle first, wheels up. I get so tired of having to pull those damn things out and flip them/turn them around to get the little door to slide shut. Many of those bags can cause the bin door to slide out of the track (because they protrude too much when placed in the bins in other ways) which then results in 30+ minute delay (longer at outstations) to get someone to come out and fix it because we can't fly with it messed up.
Also, there have been no bulletins/letters/memos about increased FAA attention to carry-on bags (or that we need to be more strict).
As always, YMMV.