I am, at best, a novice mathematician. I started teaching myself the subject while writing my thesis in computer science. I find that I have a strong urge to prove every relationship or formula that I come across while studying. I routinely find it hard to accept or understand a relationship that I cannot prove and often go to great lengths to do so. Is this practice normal or recommended or otherwise?
It is very highly recommended. Although when you begin doing things it is sometimes more fruitful to simply gaze at some ideas and accept them to let blossom more amazing ideas, at some point you just gotta start thinking and that is where foundation becomes important. It will never hurt you to start making some rigor go into your mind, as long as it does not stop it from finding new ideas. I believe that it takes much, much work to have both rigor and intuition, but that is the price to pay, because it's the best thing to do. Simply focusing on rigor won't get you anywhere though ; proving things is one thing, understanding why they're true is another.
Hope that helps,