not what the engine was designed for, either way, so yes, the gearbox will be under stress.
Less torque than an equivalent i6 engine, say.
The flat rural roads are more what the v6 would like.
Yes dirty atf will not help.
I absolutely dont know your car or the road, but at a guess youd want to be doing 3-4k revs in 2nd or 3rd and just grinding your way up the hill, not speeding up and slowing down and changing gears lots.
A decent truck should be able to do that dawn till dusk no problem. Then, you have an SUV, not a proper truck... so...