yes, you can go directly to the departures level/the gate when you have a BP for the onward flight. No need to enter Turkey to go the check-in counters.

2h45min is more than sufficient for this tranfer. You incoming flight can be late as much as 1h45min without jeopardizing the transfer.

You can not use use short connection as your transfer time is too long. Just proceed though normal transfer security checks; no issue.