Gyarados Fast TM

I wouldn't - bite allows for better dodging and I don't think dragon tail is a huge upgrade (I prefer bite). Neither gives STAB so I'd wait for possible new quick moves for gyarados before using a fast TM.

It really depends on what you are wanting him to be good at.

If you are wanting him to be a defender, then yes. If you are aiming to be an Anti Dragonite attacker, then yes. If you are like me and like to dodge, then no.

In my opinion, Bite + HP is plenty good enough and my TMs... when I get some... will go to a lot of other mons before my Bite + HP Gyarados.

No. B/HP isn't really worse than DT/HP, they both have advantages.
In fact, Bite/Crunch is no longer the crap moveset for Gyarados it once was. In fact, I would call it the best moveset now. Not because of its DPS, but its meta relevance.

No. Dark is the more desirable move for raids currently. Dragon Tail was better for defending, but no one's relying on Gyarados to be a defender anymore.

No.
Bite is SE vs Mewtwo and Gyarados resists FocusBlast.
Won't say it forever, so if you know what I mean, make profit of the idea now.

Hilariously I'm thinking about using a Fast TM on my DT/C Gyarados 84% level 29.
Will do it the day I'll swim in TMs!