Monks special attacks

And by title, I mean the abilities like Eagle Strike and Flying Kick.

Now my question regarding those abilities: which one is considered the highest DPS? Which one should I be using and maxing out first and foremost? And does that agility stay clear through to 115?

Logged into my monk (69) after years off and when setting up hot bars and abilities I noticed I had a few options to choose from and wasn’t sure which one was the go to.

Tiger Claw and Flying Kick

Are they on separate timers? I guess there’s a quick easy way to check that but I’m stuck at work for another 4 hours so may as well ask while I can. And if so, can I /autoskill both of those abilities!? That would be swell

Monks and beastlords can do two autoskills. One "strike" category attack, one "kick" category attack. All other classes are restricted to one autoskill at a time even if they have both categories available.

While Tiger Claw and Flying Kick are your bread and butter DPS - at higher levels Eagle Strike generates endurance for your entire group (6-8 step pattern as well) which in raids really can come in handy. It is situational but an important tool none the less...and the DPS on it is still quite good.

To answer your other question. Just try to get the best defiant gear you can and any type 7 augs you can find on the way and you will have enough agility and dexterity.
After level 85, try to focus on Dexterity. At high level I have 1 top of the line Agility aug, a top stamina aug and the rest dexterity (talking type 5 augs here).

Eagle strike with the AA Divine Strength of Battle has a chance to give the monk endurance and endurance regeneration. It does not hit your group.

On TLPs (PoP, GoD, for example), the best are Flying Kick and Tail Rake/Dragon Punch.

Around your level, Dragon Punch/Tail Rake would be more dps than Eagle Strike. Also use Flying Kick.

What level does tiger claw start to overtake tail rake? 85 when we get Drunken Monkey maybe? Or is there something else that boosts it?

resonant strikes AA

Outside of specific situations like the need for solid stuns (then use round kick over flying kick) I believe that tiger claw just gets more bonuses than anything else.
As for the endurance comments, as a monk you can use swift tails chant and also technique of the whatever fists for a good endurance management. The only time that endurance really becomes an issue is when our warriors die and need endurance fast, and even then I am usually asked to focus paragon or use some other BST skill that lets groups steal endurance.
I feel like my monk on full burn manages to scrape by without having to swap to eagle strike.