As I recall, the Uni Rover 100 was a pretty low-end bike--but so are the two you've mentioned as possible replacements. The Costco Schwinn is almost certain to disappoint you, I think. I don't know much about that specific bike, but I've looked at Costco bikes in general, and they're usually badly assembled, even to the point where they wouldn't be safe to ride, and if you have a problem, I don't think they'll help (Costco is good on returns, but they're not set up for bike service).
REI has a better rep, especially if your store has a bike shop (not just a sales rack), and that's a good price on the K2 (checked it on the website). You'll be comfortable on that pedals-forward frame, too (that design has a name, but I can't remember what it is...). Of the two, I'd recommend that.
Assuming there's nothing major wrong with the Rover, though, you can probably put it into decent riding shape for @ $100. A few suggestions:
New tires, maybe $25 each. Just get used to flats--they're part of cycling. With practice and a decent pump, you can whip out the old tube (to patch later) and put in a new one in about three minutes. Something like Slime (a sealant in the tube) can cut down on the numbers, but you'll never get rid of them. Buy tires to ride, not to stop bullets. And if you'll be mostly on pavement, don't buy knobbies--tell the guy in the shop you want road tires, something around 1.5 inches wide or so.
The shifting and brake adjustment problems are probably just that--adjustment problems, fixable in a few minutes by anybody who knows how. You can learn (it's not hard), or you could pay a shop a few dollars for a tune-up (I have no idea how much--I've done all my own work for years--but I'd guess less than $50). You should also pull out and lubricate (or replace) the brake and shift cables, which sounds intimidating but isn't hard. That will make a big difference in the feel. And lubricate the chain with something oily--they make chain lubes, but ordinary oil will work well enough. Flow it on, then spin the pedals backward and use a towel or rag to wipe the excess off (it will take awhile).
More complex jobs include putting in good aftermarket brake pads, but that can wait. Usually, when you resurrect a cheap old bike like that that hasn't been ridden much, the main issues are sloppy assembly to begin with coupled with a decade of no maintenance. Both are easy to fix.
As for riding hills, look at the gears on the crankset, the ones down there by the pedals. For hills, you want to shift into the SMALL gear there, and the BIG gear in the cluster of gears on the back wheel. That's your lowest gear, and it's made for hills.