I think it is possible to provide semi-generic answers to your questions
- When I should make my work available?
If your problem is interesting (say an open problem or a problem that was not been attempted or solved in the past), then you could make progress on it, write a draft and upload it to arXiv or ECCC. As long as you get endorsement and moderated by arXiv, the document will be online. Yes, when you should do it is a question? --- You should do it when you feel that the community is getting something new (or a revisit) to an existing problem. You work is expanding - to some extent - the existing knowledge about the problem or research. That should be like a visible contribution to the research in your field.
- What is the arxiv and ECCC other than that ECCC is for complexity theory and arxiv is open computer science, mathematics etc ?
As the website reads (at the bottom): arXiv is an e-print service in the fields of physics, mathematics, computer science, quantitative biology, quantitative finance, statistics, electrical engineering and systems science, and economics. Submissions to arXiv should conform to Cornell University academic standards. arXiv is owned and operated by Cornell University, a private not-for-profit educational institution. arXiv is funded by Cornell University Library, the Simons Foundation and by the member institutions.[arXiv.org]
There is a process of moderation that happens in arXiv for every paper. Read about it here.