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 how to send what 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". 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 work to someone 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!

I need to submit my thesis in a month but haven't published my results yet. Some people are hot on my trail and I'm also worried that I might "inspire" my thesis referees.

Should I submit my chapters to journals before I receive my thesis referee reports back?

Quality

Due to my previous experiences, I'm only getting people that are not directly interested in my results to proof read (this includes my supervisor) but I'm a bit worried that they are only skimming over my thesis.

Should I be concerned?

Thanks in advance for your advice :)