Is a good PhD student or researcher expected to spend some part of their day/week regularly reading new research papers/pre-prints?

There are pages like http://www.arxiv-sanity.com/ to filter the mass of papers on arXiv and give you the chance to read what interests you most. Furthermore, you can also go through all recent papers on your phone (instead of browsing instagram, reddit or whatever), read the abstract and mark everything that might be interesting to later read. I'm not sure if it is expected, that might strongly depend on your institute, group or advisor, but it is surely possible to do it in nicer ways than by simply clicking through arXiv directly.

I know good researchers who check arXiv daily, I know other just as good ones that haven't looked at it in months, so there also is no general rule there for after the PhD it seems.

Btw, if anyone knows of other nice ways to filter the arXiv feed, feel free to share/add, I'm not yet 100% happy with the ones mentioned that I currently know.

It sounds like you are not yet a graduate student. Otherwise I'd suggest that you talk to your advisor about this. In the future, if your advisor suggests that you read something, and perhaps comment on it, then you should do so. The theory is that it is always good to make your advisor happy.

But a random paper from a random person, especially another student, should put no obligation on you. Especially if you are asked for comment. But you have made yourself "public" by submitting your paper, and therefore a target for such requests. Ignore them if you like or read them if you find them interesting, and comment if you like. But there is no implied obligation.

One reason for sending out such requests is a bit sinister. The others may just be "fishing" for citations in your future work.

In a more positive light, knowing about this other work helps guide you away from problems already solved or about to be solved.

However, it is also true that graduate students do need to spend time, perhaps a lot of it, reading papers. But those papers should relate to your own research trajectory, not necessarily those of others. Later in your career you review the work of others as an implied public service to the profession, but it is too early to expect that now.