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 :)