How Much Flesh to the Bones does an Initial Online Publication need?

I personally regard an arXiv preprint as a usual article, just not refereed. One should be able to refer to it, hence it should contain full proofs. After you put your results with full proofs on arXiv, you will be able to write a short answer to your question (not necessarily as a CW) with a reference to your preprint. This acceptable; see this answer.

I presume that with "online" publication you have arXiv in mind. (For a journal it does not really make a difference whether it is online or not.) With regards to your question, the key difference between arXiv and a journal is not so much the absence of a refereeing process, but the fact that on arXiv you can post multiple revised versions. With a journal publication you get a chance to revise it after the referee reports come in, but once it's accepted no further revisions are possible.

So yes, you can submit incomplete papers to arXiv, and complete them later. The arXiv moderators will hold you to a certain standard, but if you are not submitting a proof of the Riemann hypothesis you should be OK.

There are risks and benefits with an early submission to arXiv. One benefit is that you can protect your priority. Another benefit is that you might receive helpful feedback on your findings. One risk is that a sloppy first version may damage your reputation a professional researcher. Another risk is that someone may build on your results (with proper attribution, of course), perhaps generalizing them or applying them in a different context, while you may have wanted to do that yourself.