By server rules, you are allowed to down and restrain anyone you want. Just remember to initiate with hostile RP before dowing anyone. If you're talking about other people have their bounty then yes, other people can have the same guy you're looking for and they can also down and restrain them.
When you down a person you have to zip ties, when they're downed or surrendered you can scroll on them and restrain them.
You escort on the suspect that you have detained, press escort and scroll wheel on the car and press "Put in vehicle"
To send a bounty to jail you would need to go up to a courthouse and press send to jail, watch out for other people attempting to claim the bounty as others can run up and press "send to jail" when they are by the vicinity of the courthouse. There's also an option to jail people at skip tracer base, go up to one of the NPCs and place your bounty next to them and press "Send to jail"
Hope this helps and welcome to asylum!