nothing wrong with dead hangs, I do them too, but get bored quickly. doing only pushups or any one dimensional move is a recipe for stiffness and injuries. take a look at this video, a few foundational excercises, can make progressions from very beginner level, for any age and condition and you'll find on youtube each excercise in more detail, pick what you like, disregard the rest. https://www.youtube.com/watch?v=rDh26Cu_PqQ