I can't answer all of those questions, but here are a couple of comments:
1. The Engineering PhD program ranks 37th (see the web site for just whose ranking that is). That doesn't say much about the undergraduate program in general, and it doesn't say much about the specific majors you are interested in. I think I read that the BME rank is pretty high, but I don't remember any particulars at all.
2. You can change majors. You can change schools altogether. Of course, the requirements will be different for the different fields, so if you switch later, rather than sooner, you might lose some time.
3. Some students do master's degrees in 5, some do it in 4 if they enter with a whole lot of AP credit. The two I know best who are doing it in 4 are both computer science, though, not strictly engineering, even though they are in the Engineering school. I don't know exactly how formal the program is, but it does require applying during junior year.
4. There is no co-op program. Most co-op programs end up requiring at least one additional semester to finish the bachelor's degree. My son had no difficulty finding research work for the summers, although that is because he did research t/o the school year. He found summer internships on his own, but there is a placement office to help. I don't know too much about the recruitment office; son has had his job offer in hand since before senior year, through a summer internship. The web site does provide some information about job fair/recruitment days.
5. As far as I know, there aren't any restrictions against double majors within engineering, but how easy it is to do will depend on the majors. Double majors outside of engineering are not uncommon. Many students have an additional math major.
6. Reputation--very good in the south, increasingly good everywhere else. Son's job is in CA. When you submit your resume, they look at your course work, your research experience and your letters of recommendation. They don't pull out a map and check on the geography, as far as I know.
Hopefully you will get some feedback from current or recent students who can give more particulars about the majors you are interested in. CS is the only major that is not awarded a B.E. (it is a B.S), so some of the rules/regs are different.
A general comment on VUSE. I think the relatively small size of VUSE works to the benefit of undergraduates, for the most part. The administration seems very dedicated to undergraduate education, and they have been very flexible for my son. Many of the larger programs can be pretty impersonal, inflexible and designed to maximize opportunities for graduate students, not undergraduates. One potential negative for small programs is the relative lack of breadth in the type of research available for hands-on experience. If there is a particular area that interests you at this point, it would be worthwhile to look at what kind of research is going on. Spend some time reading the web pages of faculty members, and look at what the allied research groups are involved in. Actually, even if you don't have specific interests at this time, it is worthwhile for prospective applicants to spend some time looking at faculty pages (at all schools, not just VUSE).