My shoulder pops when I do lat pulldowns and hurts when I do push ups?

Arthritis/bursitis. Possibly arthritis/bursitis. See your pcp/orthopedic doctor.
See a shoulder doc. While painless popping can be nothing to be concerned about, pain with normal function, like a push-up, requires evaluation. An orthopedic surgeon with experience with shoulder problems would be best to evaluate your shoulder.