Both as an author but also as a referee I made the experience that journals are usually not very happy to have series of papers. In particular, if it is not really clear whether part 56 still has some significant new content or if it is just incremental. I have been asked to check manuscripts precisely for that and decline them if so (on request by by some journals...)
In some sense, I can also understand this attitude: for a possible reader it is very discouraging if s/he gets part 33 into his/her hands and if s/he has the impression that s/he has to go through part 1-32, too, just to understand a little lemma in part 33. This is of course an extreme, but usually people are happy if they can read a rather short paper which has a single clear result. Of course, 99% of the papers will build on some previous resources, but I think it is a good idea to write papers rather self-contained: at least, I've been told by many people that they personally would prefer this.
Now to your problem, which is not at all unfamiliar to me ;) If you have a long string of thought, ranging from some pure/theoretical foundations to some applied examples, things become tricky. There might be no good solution, but I think even in this case, it might be good to split things into parts which can be read rather independently of each other. Then there is no need to call part 55 part 55 but it will be a separate paper which can be submitted to the optimal journal for this piece of math. Privately, you can still call it part 55 ;)
What might be an option is that after the whole project has been published in variuos journals: write a good review of it. I think, for long (over many years) projects it will become increasingly important to have good reviews by people really working in this field. Then a reader can decide whether s/he will really have to take a look at all relevant parts (papers) or just at a few ones. Of course, producing a good review is not at all an easy task, it is much more than just cut-copy-paste some pieces of previous research articles.
Hmmm, I think I have not really answered the question, but I gues what I said matches perhaps best to your point 2.). However, I really would avoid to call the title of the paper Theory XY, part IIX...
Best Answer
Put both papers in the arxiv. Have the second cite the first, if it depends on the first. You may have the first cite the second as a "preprint" as a motivation, if it's useful. This can be updated by re-posting the first paper once you know the "coordinates" (arxiv number) of the second.
If the papers form a clear sequence, you may wish to submit both papers together to the same journal. Explain why you are submitting two papers and their relation to each other in the cover letter. Otherwise submit them separately.