I'd agree with the suggestion that you use bar-ends for the front-lifting problem. There is often quite a difficult trade-off on steep off-road climbs between losing traction on the rear and lifting the front of the bike. I find that the long bar-ends on my MTB allow me to stretch out and find the right balance.
The back pain is due to the fact that those muscles are transmitting all of the force between your legs and pelvis and the rest of your body. They will get stronger as you stress them and they will stop hurting.
It's true that hard pedalling tends to rock the pelvis and the core and back muscles are having to resist that. Cycling alone may not make those muscles strong enough though - I've been cycling as an adult for 22 years and my back still aches on every ride!
Part of my problem is a lack of flexibility and cycling actually makes that worse because so much of the time is spent in an unnatural bent-over position. I keep meaning to try yoga to improve my flexibility. Several of my cycling friends swear by it.
Lack of core strength is another issue and I haven't found that cycling completely solves that problem either, though admittedly I haven't been cycling consistently enough over the past 3 or 4 years to really build up that strength. My back problem wasn't as bad when I was riding 500+ miles a month. More riding, yoga, Pilates or some other exercise to strengthen the core would help.
Carrying excessive weight doesn't help because it's an extra strain on the body. When I was 3 stone lighter, riding felt so much easier. If you can reduce the weight you are carrying on you, the bike itself or as luggage, that would help.
The other possible factor (which applies in my case and is probably the cause of 75% of my back problem) is that you might have a significant leg-length difference. My right leg is significantly longer than my left and it causes my pelvis to rock as I pedal because my left leg is struggling to reach down to the bottom of the pedal stroke. Lowering the saddle makes it feel too low for my right leg!
I'm currently experimenting with using a shorter left crank to see if that helps. When I used Look pedals, I put shims under the left cleat but that's not something that would suit the SPD cleats and mountain bike shoes that I currently use.
Low climbing gears are kinder on my back. A lot of the time I can't actually use the strength my legs have because of aches and pains elsewhere so I use very low gears and spin my way up the steep stuff. If you go too low though, it is easy to spin the back wheel so you have to pedal smoothly and not 'mash' the pedals round.