I have a strong passion to learn both UX and web development, but on top of school it seems like a burdensome task. I am studying computer science.
With so much going on, I think you need to narrow what are very broad fields and focus on certain topics within them.
"Learning UX design" is not something that actually defines your goals very well. UX practitioners themselves will argue (to little effect) what UX design actually consists of (See image below). I suggest you find a topic from the image below that sounds like it might be of interest and start by focusing on that, so that:
I can't really speak for web development, but a similar approach might also work here.
I would also add that knowing more about UX is an excellent grounding not just for web development but development of software, products and services generally as they all involve people!
For example, reading The Universal Principles of Design allows you to dip in to many different design principles that provide a great foundation for designing and creating pretty much anything.
If you want to stay abreast of what's hot, check out the programme for popular ux and web conferences, looking for talks that sound interesting to you and just read around and investigate them. And for past conferences, videos are very often available.
Some say it takes 10 years (or 10,000hrs) of deliberative practice to develop expertise in any area.
Others say that some (usability) testing is infinitely better than none.
Becoming an expert in both fields simultaneously could delay both. However, if your priority is to be an engineer, pick up a few UX books to get the concepts. It will make you more well rounded and a better programmer.
Definitely, yes! If your curriculum allows, you can fill in "gaps" by taking up classes in Industrial Design, or Psychology, or even choose a minor in one of those fields. This way, you can combine "official school time" with your own study time. UX is a multidisciplinary field, you would profit a lot from picking things up alongside your main study.
Here's a popular blogpost about "fattening up your T": good to have a speciality, but in UX, you need to broaden as well: http://markdotto.com/2011/04/15/fatten-up-those-ts/
Moreover, you can also consider to do one of those educational programs that focus completely on UX's multidisciplinarity, such as: http://www.3tu.nl/sai/en/programmes-and-tracks/usi/ -- So you do not necessarily have to do it "on top" of school: it could just be your school-time!