1. *deleted erroneous information* If you change from 3*7 to 3*8 you have two new problems: you need a new shifter to handle 8 gears and the width of the 8 speed cassette is wider than the 7 speed. The chain may rub on *deleted bad information*.
2. Any replacement derailleur must also be able to handle the largest cog on that rear derailleur, so if you are changing the rear cassette to give you a lower gear (more teeth on the largest cog), the derailleur must say it can handle it. If it says it will handle 30T, it may handle 28T well but 30T performance may not be excellent.
3. In general, there is a difference in frames from one price level to another. Entry level (low price) frames come with entry level components. If you want the more expensive components that usually come on a more expensive bike, it may be better to simply replace the broken part with the same item originally installed, and sell the bike. Then you can get a bike with nicer components and you will have a nicer (lighter perhaps?) frame also. It all begins with the frame. In automobile terms, you can put expensive rims and tires on a Chevrolet, but it will still be a Chevrolet. Buy a nicer car to begin with; it will probably come with the nicer rims and tires that you want and the car will be a nicer car, also. But that is only my opinion. I see a bunch of entry level cars around here with rims and tires that cost as much as the car itself. It doesn't make sense to me, but lots of people do it. Good Luck!