Will these new wheels be able to replace the old ones?
The frame will take any 700c 135mm hybrid wheel, which can be either freewheel or cassette. You likely have a freewheel hub, which means you have a freewheel, which means if you buy a cassette wheel like you are talking about you will need a 7-speed cassette plus a 4.5mm spacer to go with it. And then you're putting a new cassette on a worn chain and chainrings, which could turn into needing to replace at least the chain and perhaps all of it, depending on how worn it is, which may seem cautionary but we're also talking about a bike that was heavily used enough to be riding around with a broken axle.
You could look for a used hybrid freewheel wheel, but without knowing what you're looking at, it would be easy to come up with one that's in poor shape. And, it sounds like you have no particular reason to replace the front anyway. The 18 vs 19 internal width doesn't matter.
7-speed cassette hubs almost never break axles, but if you did have one and the wheelset you're looking at was a good deal, then yes it would be fine.
Unless a project is what you want, there is a lot to be said for just having the shop install a new freewheel wheel and adjust everything for you.
If spending as little as possible is the goal and you're trying to do it all yourself, it is possible you could spend the least by replacing the axle. It's common for broken axles to result in the cups and cones getting too damaged for a shop to want to mess around with, but cheap axles are cheap and you may be able to get it rolling. If it wasn't ridden too much after breaking, it may even be fine, and it's falling apart anyway so it's easy to look at the condition of the cones. You would need a cone wrench and freewheel tool plus grease and bearings to do this.