Should I worry that my 4-month-old does not do push-ups?

No. Nowadays, babies sleep on their backs. They don't have much opportunity to practice pushing up, so many or most 4-month old babies can't do push-ups yet. (many years ago, when babies slept on their tummies, many babies were strong enough to push up and raise their chests off the beds by 4 months of age.).
No. Babies all develop in their own unique way. With so much time spent upon their backs, " pushups " are sometimes seen later in age than it used to be. Tummy time helps promote pushups, but your baby will be fine no matter.
No. I assume you mean in the head up off the floor pushup, not an army one. No, some children dislike being on their bellies so much they just scream. Or they're content and just look around. By 4-6 mo they should be able to do head and shoulder push off the floor, but that and then rolling is a 6-8 mo skill, not a 4 mo. There should be progress though, and if still not pushing up well by 6mo, call.