[rweba]

(1) I don't think this is necessarily the right place to ask this. HN is not particularly known for mathematics, it's better to ask at a math focused site like https://math.stackexchange.com/

(2) My generic answer to the question is - NO. You can be a good quant, engineer, etc. without having the knowledge of how to prove the theorems in a real analysis undergraduate course. However, knowing the proofs may be helpful if you decide you want to do write theoretical research papers in a particular area, even just for the sake of "mathematical maturity".