I got some lower back pain 5 months ago when squatting heavy (with bad form) as well as doing weighted pullups and relaxing my back. In particular, relaxing my back with weights started the back pain; I guess the pulling force of the weights was too much for my spine. I have avoided squats and made sure to keep my back tight when doing weighted pullups (my back feels fine if I keep it tight), and I thought I recovered but last week I started doing squats and the pain came back. I kept good form, did not round my back, and even had a lifting belt, but I think my back cannot handle compression forces very well. Is there any way I can train for this, or do I need more rest? Are there any back exercises I can do?
Drop the weight on your squats and learn how to breath properly. Before each rep take a sharp breath of air pushing your stomach out, not in and hold it, contract and tighten your abdominals. Hold this throughout the whole rep. Only breath out once the rep is finished. This should help brace your spine and stop your torso from collapsing under heavy loads. I say this assuming your form is good. If the pain persists I'd recommend a physio.
It looks like all your attention is on your upper body and lower body exercises, try developing the core muscles as well, developing your core will increase balance/stability and will reduce the chances of injury. Try Planks(All variations), crunches and back extensions. Running/jogging also builds up your core. If the chronic pain is too much or is consistently there even after exercises, then you might as well get diagnosed by a doctor.
Sounds like an imbalance of core to back strength, aka gods weight belt.
Basically when doing any kind of multi joint or compound exercise, you need to think about engaging the core because it is the anterior wall to the lower back.
What happens during a squat, when someone does not engage the core is that the back and quads over-compensates for the movement and the hamstrings and abs get weaker.
Think about how much load/force is being put onto the back if you completely relax your core.
Over many years the back becomes tight and overworked and eventually becomes prone to injury.
Learn to balance your body, focus on front lever, body lever, leg lifts, saxon side bends, to strengthen your core.