1. Yes, recovering creates antibodies, which provide immunity (immunity = the person's immune system will quickly defeat subsequent exposures). There may be some variation in the degree of immunity related to the severity of the original infection (much as with chicken pox).
2. There are quite a few FDA-approved tests that can tell whether a person has antibodies to SARS-CoV-2. I don't know, at present, whether any can discern the concentration of antibodies.
3. The reason we do not know how long natural and vaccination immunities last is that SARS-CoV-2 has only been in circulation in the US a bit over a year (there is evidence from antibodies being found in blood from December 2019 blood drives that it may have been brought into the US in November). The people infected or vaccinated (Phase 1 test participants) early still have significant antibody concentrations is known.
4. "Variants" borders on Panic Porn. Viruses mutate readily. If one goes to an extreme level of detail, there are probably millions or billions of sub-sub-sub-sub-sub-sub-sub-sub-sub-sub-sub-sub-sub-sub-sub-sub- ... -sub-sub-sub-sub-variants "out there". What is known is that of the five vaccines the US government worked with, Dynavax's (not yet approved), Johnson & Johnson's, Moderna's, and Pfizer's are still effective against the major known variants. AstraZeneca's, apparently, is not effective in preventing infections against the "South African" variant (it may, however, be effective in reducing serious and severe infections).
5. Because of the mRNA technology Moderna uses, they had their vaccine designed in less than a week after the DNA sequence of the SARS-CoV-2 was leaked from China. Moderna began Phase 1 (i.e., human paricipants) testing in mid March. That Phase 1 testing of Pfizer's vaccine began in late April suggests to me that BioNTech also was able to develop their mRNA vaccine similarly quickly. AstraZeneca also began Phase 1 testing in late April. I could post similar timeline info for Dynavax and J & J, but the point is that effective vaccines were actually developed and gotten into test very quickly.