Is it really safe to do pushups every day??

yea man you will be fine

damn all those years people got smoked every day in military training and schools

I don't think you need rest but if your form isn't correct you can frick your shoulders up.

When I was in high school I was doing pushups every day but I wasn't doing sets, just seeing how many I could do in a row, working my way up to 100. I got to 88 one day and the next day I couldn't do 15. Even weeks later I was never able to do more than 20 without pain in my shoulder joints.

I started doing them again when the pushup challenge thread popped up but I'm limiting myself to sets of 10 and keeping my hands well below my shoulders.