If you pedal at 60rpm with high watts on the flats, you'll probably get the same back pain. Maybe try a lower gearing if you're in the mountains a lot, and spin up them at 90rpm.
Yes.
But a 10% hill at 90 rpm with even a 34-29 is a tough 8.5 mph. I often use my 34-29 on a 5% grade, spinning it at 90 rpm, at the same 8.5 mph.
On steep hills, low gears will help anyway, even at low rpms. I go up short 10% grades at 5-6 mph, which is about 55-60 rpm. But with my 34-29, I can mostly do them seated without putting out an extreme effort. It lets me pedal slow enough on an 8% grade with light pedal pressure that it can be a recovery zone on a steeper climb.
I did get temporary low back pain at the top of a 2 mile, 10% average climb with 18% switchbacks. That was standing and grinding up most of the way. It's the "push pedals as hard as possible to keep from falling over" where I get the back pain. (But short steep pitches don't give me any back pain)
If you are getting the pain regularly, a triple might be worth it, so you can keep doing the climbs with faster cadence and less pedal force.
And core exercises are always good anyway. One of those blow up stability balls was recommended to me, and I think it helped. But I'm just not very good at getting around to non-bike exercise.