Note, however, that accelerators generally require the entire team to be 100% dedicated to their startup. If you have another job, and/or if you have no co-founders, your chances of being accepted to an accelerator are slim to none (some even expressly forbid it).
I've never heard of a VC that suggests you apply to an accelerator, but it sounds like they were just trying to get rid of you without saying no (VCs hate saying no). Check if that representative you mentioned was a GP (general partner) or an "associate partner" (I put my money on the second option). Unlike GPs, associates decide nothing - talking to them is a waste of time at best.
4. There is no solid answer to these questions, even though entire books were written on the subject. Funding depends on many factors, and everything changes constantly. At present, people usually talk about 5 stages: pre-seed (idea or early development), seed (minimal viable product, first customers), round A (growing revenues), round B (small company), and round C (small-medium company).
Accelerators usually take pre-seed startups and turn them into seed or round A. VCs usually don't come in until round A.
5-6. Early stage investors (angels, micro-VCs) often provide a fixed investment (or at least in some narrow range). For example, they'll invest $200K for 20% (which values your startup at $1M), no matter who or what. Late stage investors (most VCs) usually won't invest at all if you don't have revenues. So overall, early stage valuation is not really a big issue.
8. A good accelerator pays for itself. In any case, the money they invest is so small it doesn't really matter - accelerator founders could pay it by themselves, if they wanted to.
9. YCombinator proved that accelerators are a great business in their own right - that's generally enough motivation. Some accelerators are supported by large companies (Microsoft has a major one) - I guess it allows them to scout for big talents in hopes of hiring them later. As for the mentors... well, some of them might get paid for it (a job is a job), but I think most see it as a way of a) helping young entrepreneurs like the ones they used to be, b) putting it on their resumes (it does look good), and c) plain old ego boosting.
Hope that didn't sound too cynical.