I would print out this from the TSA and carry it with you on both ends of the flight:
TSA: Traveling with Special Items
Your flies "should" be OK, but remember that these are TSA "Guidelines" and an individual air carrier may choose to have more restrictive policies--- or may have folks that are unfamiliar with the latest interpretations.
That said, I would NOT plan on carrying on fly tying tools or a vise on board, and would definitely not plan on tying on the flight---- after your fellow passengers potentially wrestle you to the ground, you could end up on the TSA's "s--- list" and be subjected to cavity searches every time you fly...
I would plan on carrying on your reels, camera, any 9' or shorter rods that break down to four pieces in a sturdy metal or pvc tube, and your 1/2 of your fishing flies.
I would pack the other 1/2 of your flies (just in case the carry on flies get confiscated even though they should be OK), and any fishing tools (hook file, pliers, knife, Boga etc), tying stuff (vise, tools, any bare hooks, materials) and any goopy stuff (sunscreen, zinc oxide etc) in heavy duty freezer bags to prevent spillage) as well as any rods that break down longer than 30" sections in heavy duty tubes (aluminum or heavy PVC).
Good luck--- have a great trip.