How should I publish my "story"? i.e. strategies and advice on chopping up a thesis Background
In my thesis I look at same problem from a couple of different angles. To state it roughly, in each chapter I use a different technique or area of mathematics to try and gain further insight into a hard case that was shown to have a "negative" result in the early 90s. I believe my main contribution is a few cute insights/tricks and building connections between a couple of different areas/problems in mathematics.
Chopping it
I think I have about about 4-5 publishable chapters in my thesis but they all consider the same problem.

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?

My topic is 'probabilistic' and I think my results are novel. Unfortunately, I don't really know where to submit. Some related problems and theory have previously appeared in the Annals of Probability and Journal of Functional Analysis. Although novel, I don't really know if my work is of that technical calibre though. What other journals should I consider? Should I post on ArXiv first?
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?
Timing
I think my ideas have a lot of potential and would like to submit to a top journal, but due to timing constraints and working by myself, I still think my work is a bit "shallow" (i.e. easy to extend). Ideally, I would like to have a good co-author to refine and deepen the ideas and turn it into a great publication but every time I mention my example problem it seems like I inspire them and shortly after they produce a paper with their co-authors on a similar or more abstract result. Very frustrating as I don't have the time to do these extensions myself at the moment!
I'm aiming to submit my thesis in about a month, then I will spend a couple of weeks putting the chapters into a publication format for submission to arXiv or a journal.

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

Thanks in advance for your advice :)
 A: 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.
A: Posting on arxiv is generally considered a good idea if you are worried about competition from similar results. On the other hand, you will give other mathematicians the means to adapt your ideas and possibly prove better, and possibly more abstract results. But hey that's par for the course, and you cannot (or should not want to) hold onto your results forever.
As to where should you submit, I don't think MO users can really tell from the info included in your question. Of course, you should look at journals that have published similar stuff, but isn't this where your supervisor should really come in?
[Added Later: It's not unusual to work with a supervisor who is not a specialist on your topic. It's can be valuable, since it forces you to learn autonomy in research a lot earlier. But whatever your supervisor specialty is, they should be able to give you good advice on the non-technical aspects of your endeavor, including how to write, submit, career advice and so on.]
