Little late here but I thought of a few things:
Having strong (not just big/pumped up) glutes and hams usually benefits a bad back.
So, in that case, I’d recommend doing some glute/hip activation every day. My buddy (who had pretty bad back issues) did this and it helped:
A. Bodyweight (BW) Glute Bridges: 10 reps w/ 5 sec. squeeze at the top
B. BW Bird Dogs: 10 reps w/ 5 sec. hold at the top (do all 10 reps on one leg, then do the other side)
C1. Lying Leg Abduction: 20 reps
C2. Clams: 20 reps (do 20 reps of C1, followed by 20 reps of C2, all on one leg before switching and doing the other leg)
Rest 1 min. and then do the circuit one more time. Shouldn’t take too long.
Just seems like normally people’s glutes and abductors are weak, under-used and under-worked, and not activating properly. If you can do these, or similar movements, do this every day. Maybe right when you wake up. Don’t worry about adding weight or making it really hard, just try to really feel these muscles and get them working.
If possible, do more hip extension movements. Lying Ham Curls are a great movement for pumping up the muscle, but train knee flexion, one of the 2 movements the hamstring does.
RDL’s, Good-Mornings, and Hip Thrusts are the main strength builders I can think of…SL deadlifts are good but they’re quite similar to a normal deadlift and need to be done properly if you don’t want the lower back to take over.
RDL’s can be done very slowly, with very light dumbbells, and will work the muscle while allowing you to focus on the hamstrings/glutes taking over. Good-Mornings can be done with a band around your neck, and serve the same purpose - slow moving, paying attention to the right muscles.
Hip Thrusts, Glute Bridges, and all single-leg, elevated, or whatever variations you can think of for that general movement all train the glutes and hamstrings very well. Make sure if you do them you keep your chin tucked hard into your chest, staring at the wall in front of you, especially on the way up/at the top. This prevents overextension on your lower back at the top.
I’d really recommend GHR’s if you can handle them…they’re quite hard if you’re a beginner at them, but train the hamstring from both ends of the muscle (hip and knee), and are a fantastic strength builder. Probably in the top movements for the hamstrings.
Back Extensions are good too, but if you have weak hams/glutes, you will feel them primarily in your lower back, and the constant flexion of the back may bother you. I used to feel them all in my lower back, then my hams/glutes got stronger and took over, and now they’re one of my favorite movements for those muscles.
Hack Squats and Lunges are good for the quads and a little bit of the glutes. If your knees feel fine, go ahead with them. If you get your hamstrings strong as f**k then your knees should be ready for just about anything.
I have no calves, so I won’t comment on that too much, but I’d say do standing calf raises, seated calf raises, and donkey raises, 4-6 days a week. Hit them hard and often, with as many variations as you can. This has resulted in the little growth I’ve seen with mine. (So has gaining weight lol.)
Finally, sled work has been good for me, and others I know with back pain. You can hit just about every muscle in your lower body (and upper body), cause little soreness due to the nature of it, raise your work capacity and get in better shape. (Backwards sled drags are supposed to be good for your VMO, which is hard to hit with most movements. Haven’t done it enough to report on that, but I do feel it there when I do them.)
I’m not an expert by any means, but basically, I’d recommend activating your glutes every single day, doing whatever possible movements to actually strengthen your glutes and hamstrings (this will do an amazing job at protecting your back and knees when done well), hitting your calves more often if that’s important to you, and maybe getting out the sled to work your legs and your conditioning at the same time. Here’s an article for that: Louie Simmons - GPP
Hope it goes well!