No contest: Work on the product idea and forget about the MBA.
Most of what you need to learn about the business aspects of a start up is pretty easy to learn especially for someone who has been through an engineering program. After reading a few books on the topic (such as High Tech Startup and Portable MBA series) you'll know more than enough of what you need to learn about business (relative to what you would have gotten from an MBA program anyway). For the most part, the only form of business experience that applies to startups is having done one already anyway. I talk about this to some extent in this answer to another question on this forum.
Usually, for engineers the hard part is not learning the business stuff, its actually wanting to do it over the technical stuff. Having the MBA is not going to fix that if your inclination is toward the technical part longer term anyway.
In specific answers to your questions:
1) Having an MBA does not make you more likely to get funded. This is about the quality of your business concept, the associated market it is in, their sense of your dedication to the concept and the ability of you and the team you put together to execute it.
2) Does not apply - I am not recommending the MBA path. However, if you do have an MBA I do think the quality of the program actually does count. Rightly or wrongly, from a perception standpoint, an MBA from a community college is not the same thing as an MBA from Harvard. But again, in general, having an MBA is not really going to convince a VC to fund you anyway. This is about ability to execute, not education.
3) Yes - work on the product and pickup the business concepts on the side. And, when the time comes, if you really need the help it should be comparatively easy to partner with somebody who knows how to sell what you made.
Somewhat related to your question here, I recently saw a presentation by Aaron Patzer of mint.com. Aaron recently sold his business to Intuit for $170M dollars. For a pre-revenue company, Aaron suggests that a tech company's value should be multiplied by $500,000 for each engineer on the team and you should subtract $250,000 for each business person/MBA working in the company at that point. The reasoning is simple, engineers build value in the company and the business people are only burning money at that stage because you have nothing to sell yet. I could explain further, but you should just checkout the presentation and watch the video as Aaron can explain it better.