1) Is it possible to leave the baseboards on? The kitchen has quarter-round but the LR does not.
It's possible, but would depend on how high the baseboard is setup off the floor, and what thickness of laminate you are installing. If you have enough clearance, it could be as simple as removing all the quarter-round and reinstalling it after the laminate install, this would require you to place quarter-round where you previously didn't have it. Remember that you need to leave 1/4" expansion on all sides for expansion.
2) In the kitchen, we have a couple of dips where we can see/feel the vinyl move when we step on it. What's the best way to determine if it is level enough to lay on top of it? If it is not, will I need to pull up all the vinyl? How big of a job is that?
If you can see and feel the dips in the kitchen then you will need to address this issue before you do any laminate work. If the flooring is moving, it will cause lots of problems on a laminate floor, being that the laminate doesn't have a secure sub-structure. This might require you adding blocking under the kitchen floor inbetween joists or adding another layer of plywood to the top (the second option is what I would recommend).
3) What's the best kind of saw to use to cut the laminate? I have read several different suggestions.
More important than the kind of saw, would be what type of blade to use. You will want a fine toothed blade for finish work. The type of saw depends on what you are cutting, if you are just cross cutting laminate I would use my mitre saw, if ripping a board use a table saw or a circular saw with a straight edge, and finally if you are cutting out for a floor plug or any type of non-straight cut I would use a jigsaw with a finish blade (bosch makes really good finish blades, that cause very little chipout).
Hope that helps and good luck with your project.