Hows your bike set up? My back used to ache (not hurt) when i first started cycling for a decent length of time even though id been deadlifting 550lb and squatting 500lb lol.
I think its just about being sensible. If squats hurt, then try a different bar position, different stance or footwear and include some stretching. If they still hurt - sub them for leg presses
leg presses are cool but i dont have a leg press machine and i do love squatting (probably cause im actually ok at it).
As for core work, i do 1-2 exercises per session. Lower back, abs and obliques, & rotate them. Deadlifts, and hamstring work should be productive too. However Hamstrings could be the source from the back pain - if theyre tight just ensure you stretch them after a session, or when your watching tv on a night or something.