Are they compatible?

eg: Can Hal Jordan move half his movement and TK a teammate.

or can he running shot adjacent to an enemy and use option 2 to TK an opponent off a building.

or can he RS next to an object and do a TK attack with that object?

My first instinct is yes, and i have been laying that way but is it legal?

the second eg I ould think not because that seems like a close combat action to me. So could a charging TKer (is there one, can't think of one right now) do that?

i'm pretty sure that you can RS and tk an object as running shot gives you a range combat action and when you tk an object to hit somebody with it's a range combat action so as far as i know you can only do number 3 hope that was helpful!

You can't.
Using TK, in any way, requires a Power Action. Running Shot only gives you a ranged combat action, so they are not compatible.
Also, FYI, you cannot damage an enemy by TK-ing them. You can move them all you want, but they won't take any knockback damage. For that, you need to Force Blast them or just cause regular knockback.
And BTW, U Kilowog has Charge and TK...