If your body is doing strange things, you can see/call a doctor, physician's assistant, orthopedic surgeon, physiotherapist or other health care professional, research on a reputable medical website for causes of joint popping and self-evaluate, ignore it in the hopes it will go away or pursue a number of other options.
However, to base your well-being on the word of anonymous Internet posters is perhaps not your best course of action. While there are medical professionals that post on this site, Army.ca is neither an accredited nor authoritative source of medical information.
As for your second question, there are several topics concerning physical training in general and push ups in particular. A search will turn up a great deal of information and should serve to guide what is perhaps the overall principle for self-motivated PT training:
"Do what works for you, does not injure you and helps you achieve the results you seek."
Good luck in your search and let's limit the replies to those that offer some value.
The Milnet.ca Staff