I’ll answer Question #2 first, as I think it’s easier.
What you do matters far more than where you go - and students go to top grad schools from all kinds of places. So if you chose to go to Regis, you could still go to a top grad program.
Your undergrad degree matters more indirectly than directly. Some colleges offer better opportunities to get the kinds of experiences that make you more competitive for grad school. For an extreme example, community colleges and small commuter regional schools often don’t offer great opportunities to get involved in research, as a lot of times their professors don’t have time to do research. Conversely, large prestigious places will offer lots of different, diverse ways to do research; some have things that are difficult to get elsewhere (like a large observatory or time to work with primates or whatever else).
But generally speaking as an undergrad you don’t have to worry about doing “cutting-edge” research or even exactly what you think you want to do as a grad student. You just need to learn the basics of research in your field and have a mentor who is willing to teach you. You can get that at many places.
As for question #1 - it matters a lot, but how much depends on your field.
The top schools in a field vary a lot by the field itself. But my advice is definitely to get out of the “Ivy League” mindset and realize that the top schools on the graduate/research level are much, more broader. For example, in math, top programs include Ivy League schools Princeton and Harvard; other elite private universities, like NYU, Stanford, and MIT; and top public universities, like Berkeley, Michigan, Wisconsin, and Penn State. The same is true in philosophy and neuroscience. So when you are looking for programs, don’t use your knowledge of what universities are prestigious in the undergrad sense to decide - look at rankings designed for grad programs, ask your professors, and pay attention to who’s publishing research in the fields you’re interested in.
In some fields - particularly the humanities, including philosophy - there aren’t many options for doctoral degree holders to do scholarship in the field outside of academia, and so competition for tenure-track academic jobs is fierce. In those fields, going to a prestigious place is very important when it comes to getting jobs. (But again, prestige is relative; Rutgers has one of the top programs in philosophy.)
In other fields - like engineering, and applied math - PhD holders can make a lot more money outside of the academy, and there are lots of options for someone who wants to do research in their field to do it in jobs that aren’t professorships. So competition for professor positions in those fields is less fierce. While going to a prestigious program does definitely make a difference, it’s also easier for people who have gone to more mid-tier programs to get tenure-track positions.