Note that I am not currently working for an engineering firm, so take my advice with a grain of salt. I have, however, recently completed a computer science degree at a top-10 U.S. university.
Does reputation really matter, and how much will it affect my salary?
A lot of top employers will gravitate toward more renowned schools and so it makes connecting easier and in perception, at least, more organic. With regards to salary, it only affects the job offer. Once you have shown ability through work, duration of experience and quality of work matter 10:1. At the undergraduate level, education quality and reputation are somewhat inelastic, but it depends on the school. It's at the masters and PhD level especially where reputation is the most important factor. Something to keep in mind if you plan on submatriculating.
Is transferring to a different school a good idea?
You should go somewhere you are happy, honestly. If you share some common interests with your peers, you will be much better off immediately. However, if the curriculum is too easy for you and is not offering you sufficient opportunites to hone your skills and develop your weaknesses, then it may be a good idea to look elsewhere. You want to be somewhere with a tangible energy on campus and a sense of community. If your school is lacking this, you might not be able to connect with others and should transfer.
How much can getting a masters/PhD in the field benefit me?
This is a lot less straightforward than most imagine. One thing to consider is that a lot of top students are opting for masters in droves. As it is usually only 1 year of additional schoolwork (for which you can receive financial aid if you submatriculate), I would certainly consider it and do some preliminary planning once you are settled in. Master's degrees are desirable to employers, especially in engineering fields and boost the starting salary by a considerable margin.
A PhD, however, is a much different road. It is wholly research based, so you should only consider it if you love research and wouldn't mind doing nothing but it for years straight, making little money. You'll need to prove yourself with your name on a paper or two during undergraduate in order to entice the top institutions into funding you, as reputation is all that matters. That being said, engineering PhDs are the most valuable of all the potential PhDs to employers, and I argue that if you are passionate about computer science and have a talent for it (namely math), then a CS PhD just might be the most valuable thing you can get from universities for the future economy. If it's in anything else, be aware that too many employers, unfortunately, see a PhD as somewhat unemployable because they find a 30 year old without private sector experience to actually be a sketchy prospect. It's sad but it's true. Additionally, PhDs in the private sector do not enjoy a higher starting salary than those with master's degrees. If you want to be a professor or command a top position in Big Pharma, then it is your path to entry, but otherwise you probably shouldn't. Live life instead.
Is there any advice you have for a young engineering student like me?
(any regrets?)
Challenge yourself, but don't overwork yourself. Work smart, not hard. The worst thing you can do is avoid courses that interest you because they seem too difficult. Hardship is underrated, and you'll find a lot of students picking easier majors to avoid it (in fact I think that's a big problem behind this economy's stagnation, but that's for another time). If you can find your personal limit for a given semester, sit a half step below it and make each day count. It'll be over before you know it.
Make really good friends as soon as you can because they can remain companions for life and will be your support during the tough times (there will be some, I assure you). Despite my school having average SAT scores in the mid-700s, a disgustingly large number of undergraduates were narcissistic, unexamined, blabbering brats. They took everything for granted and felt comfortable in popular contemporary opinions. It wasn't until halfway through that I really found people like myself, and I really wish I had found them sooner.
Treat it like an enrichment experience. There may not be many more times in your life where you can spend each day in a dense information network of your peers without being expected to be wholly self-sufficient. If you're really into engineering, make everything else you do separate from it. Don't be the guy who's an engineer by day and MMO addict by night. You'll spend much of the rest of your life doing what you want with your time anyway, so take this opportunity to do things that you might normally not do.
Yeah, no, I don't know? In the scale of things the engineering school you went to isn't huge. If you know your stuff it won't matter much. 
