Caveat: I just finished my PhD, and I have spent the bulk of this semester converting my dissertation to papers. The best advice is Pete Clark's: Ask your thesis advisor for advice. Nonetheless, I'll try to answer your questions to the best of my ability.

Should I submit to arXiv or journals before I receive my thesis referee reports back? or wait until I receive their comments?

It can't hurt to stick the entire thesis on the arXiv right now, then update it when you've completely finished the thesis. If you're worried about any copyright issues, this is the place to put it.

Do I send them all to the same journal and let them take their pick? or different journals? but then how do I choose which chapter to which journal?

Absolutely not. You shouldn't publish your thesis, per se. You should write and publish good papers, which will be based on the work in your thesis.

You write a thesis because it is a formal document certifying you have earned your degree. You write papers in order to communicate ideas and methods to other mathematicians.

How do you choose which journal to submit something to? Ask your thesis advisor for advice.

If you think your work is "shallow", then it's not appropriate for a top journal. That's okay: there's no shame in submitting it to a lower-tier journal. The important thing is to get it out there ASAP. You can then spend your time and energy on doing something totally new and deep then submit that to a top journal.

My favourite result is a very short chapter where I derive some cute estimates for a special case by "bare hands" without any big machinery. It is totally non-standard estimate but shows a new type of question one might pose about my problem (and also other problems). The paper would only be about 8-10 pages long, is this worthy to submit?

It is absolutely worthy to submit. Is it worthy to be published? Well, that's between you, the referee and the editor of the journal you submit it to.

Since you have 5 chapters, here's my suggestion. Turn your favorite chapter into a publication first. Then condense three of the other chapters into a second publication. Don't publish the last chapter outright. Instead of answering the same question a fifth time, use the method you developed in the last chapter to answer a new question, then publish that.