"What is the feel of Portland like? Like in the city I currently live in, there is one downtown area, and then all other parts/neighborhoods in the city have more a suburban feel. Is Portland like this or is it more city?"
There are a lot of very distinct neighborhoods in Portland proper. Even within downtown, there's the shopping area around Pioneer Place, Old Town with all the bars/nightlife, NW 23rd with boutiques and close to the Timbers stadium. On the east side the neighborhoods tend to be more bohemian. The closer to the river you stay, the more urban it feels. There further out you go, the more suburban it gets. Lewis & Clark's campus is in a somewhat suburban area of SW Portland, surrounded by forest.
"Public transportation in Portland... does an adequate system exist or is having a car a necessity? I'm just not sure if I would be able to bring my car from the east coast."
I have a car, but I know plenty of people who get by with bike+public transit. Again, the closer you stay to the river, the easier it is to stay connected to transit. Though, if you do want to live further out, where rent is cheaper, just make sure you're walking distance to a MAX line, and you'll be fine.
"If known, where do a lot of law students live? Is it pretty spread out or are there more confined areas of law students?"
I'm 0L, so I can't really speak to that.
"What are the surroundings of the law school like, what is the area/neighborhood like? It appears to be cut off from the undergrad campus, can anyone comment on this."
The surroundings of the campus are pretty suburban, in the sense that you have to drive/take a bus to downtown. But it's not like strip malls and housing developments. It's lush forest.
"What is living in downtown like? General idea of cost?"
I looked at a lot of studios in downtown Portland. They were going for $650-$800, plus some utilities. The rental market in Portland is very tight, like 3% vacancy, compared to 5% in other cities. Best advice: make friends and rent a house together. That makes the cost more like $400 per person. Personally I prefer living on the east side, but now I'm thinking about moving to SW Portland to make the commute to L&C easier.