Most likely Costochondritis. I have this chest popping sound also, one day the cartilage inbetween my breastbone was stressed somehow (emotional? gym? don't know) and it was unbearable. I had a hard time rolling out of bed. Barely could do a pushup like you without discomfort. I went to the hospital thinking I had some serious issues. Yet they told me about the cartilage moving around in the middle.
They just gave me some motrin and told me to rest it off. I think dips were the culprit that irritated it the most. Stopped doing them and I've never had that weird shit happen again. So I suggest just take some motrin and not do anything that will stress your chest for a while. They told me it could take a couple weeks to feel better. Take this time and work on you back and legs lol.
So I suggest doing that and if it doesn't help go see a doctor lol.