I would suggest you take a week or so off from training just to let your niggles calm down, then try again. You've been doing it for a month - it is perfectly normal to feel soreness in strange places any time you start something new. You are not used to feeling pain in those area which is why you notice it more.
The elbow pain is quite possible hyper extension of the elbow joint while punching. Check this previous reply for a bit more detail.
As for your wrists, it could be just very mild tendon or muscle strain or fatigue. I'm also a software engineer so I'm well accustomed to the niggles with that profession - mostly they manifest as persistent ache in the anterior deltoid (front of the shoulder), knots in the trapezius or rhomboideus (upper back, back of shoulder into neck), or as sharper pain in the wrist/palm area. You've changed what your body is used to, and now ou're forcing your wrists and shoulders to do work - so of course you are going to notice some discomfort in these areas, especially if you're not allowing enough time between sessions for the muscles to repair themselves.
Binding your wrists better may help, although I'm a fan of learning to punch without wraps (I think it leads to better strength and technique).
