Regarding your 3 questions about your profile, in order:
- I think people with undergrad in engineering is quite common in business. I'm in Marketing and I think at least 4 of my professors here have an undergrad in engineering. I don't think this would be a problem in management either.
- Yeah, right now your profile seems very weak. Like, there are people with low GMAT scores that are accepted by great schools, but usually they are very strong in other ways. You're talking about low GPA, very low GMAT, no research experience at all, probably not great letters of recommendation. Even your reasons to do a PhD don't look good to me. Someone "enchanted by the world of management consulting" does not sound good to me for a PhD. The same goes for your "trepidation" with lifestyle and work balance issues. And you wrote that you love to learn and want to know things, but a PhD is something different. Yeah, you need to learn and know things, but the big challenge and purpose of a PhD is not that. It is discovering new things, not learning and knowing things. This all makes me feel like you still don't know enough about what you are going after.
- Going into consulting now seems like going in the wrong direction. After all, if you really want to go for a PhD, I would expect you to do something related to academia, not industry. Getting research experience would be great, getting teaching experience might be still good, but getting consulting experience? I don't think it would make such a big difference, it could actually make things worse.