OK, so I've read the rules for newbies and I'm going to try not to contravene any of them. Whilst I am a hugely keen bike rider, it has become abundantly clear to me in the last month or so that I don't have the first clue about bike repair or maintenance. Apologies if I break any of the rules here.

Here goes:

I buckled both my wheels in an accident a fortnight or so before Christmas. Being a student, I don't have much spare cash and I could only afford 1 replacement wheel. I bought the front one, and asked my Dad for a 7-speed rear wheel as my Christmas present. My previous rear wheel was of the 'traditional freewheel' variety, and of course the replacement I've been given is a freehub and casette dealio.

Basically my question is, if I pick up the necessary casette to go on my freehub, will this new wheel be compatible with my bike?

Please be gentle!

Without going into too much detail, there could be (most likely will be) a difference with the frame spacing. See Sheldon Brown's site: https://www.sheldonbrown.com/frame-spacing.html . It is possible that you could go from 126 mm to 130 mm just by "jamming" the wheel into the frame.

You'll also need a spacer for a 7 speed cassette on a 8/9 speed freehub, so while you're over at Sheldon's page peruse this one https://www.sheldonbrown.com/speeds.html.

The rest of the bike is okay but both wheels got trashed? Is there a good story? Hopefully with a happy ending...

A hint for posting, providing as much information as possible is a good thing (like what kind of bike, names/models of components, that sort of thing). Mindreading is hard.

You need a 4.5mm spacer (they make those specifically for your application, a 7 sp cassette on a 8/9/10 freehub. So yes your new rear wheel will work. If your chain is at all any worn, you may encounter skipping with a new cassette in which case you will need a new chain as well.

Can you return the gift wheel and use the cash for the "correct" replacement?

+1
That would be the cost effective choice.

Especially if you have an aluminum frame, because you can't spread those to accept a wider hub.

Also look on Sheldon site for alternate routing of gearshift cabel to make 7 sp cog spacing work with 6 sp shifter. Then when you have more money you could buy 7sp shifters. Keep the 7 sp freehub wheel because the hub is much less susceptible to axle breakage.

You can find many articles related to this situation here:

https://sheldonbrown.com/gearing/index.html

Personally I'd stockpile the 7-speed parts. They are getting scarce.

+ 1 Billion. Preferably bringing the bike with you and leaving it with them until it's repaired.

You can, just not permanently "cold set" aluminum frames.

Is the new one for a seven speed cassette or for an eight? A seven will have the same rear spacing as your freewheel hub.

Very true. I wouldn't try to put a 135mm hub in a 126mm aluminum frame but 126 in a 130 has never given me any problems (Cannondale road frames from the late 80's in both cases).

Not necessarily. I've worked on a few MTBs from the '90s with 7 speed cassettes and 135mm rear spacing.

I have not tried with an aluminum frame but respaced a steel Schwinn from 126 to 130. The hub would not stay put until I bent the dropouts inward to bear squarely on the ends of the hub, unless I used too much skewer tension, which is a recipe for ruined cones.

I am glad it worked for some people but since I follow conventional advice about how tight to run the skewer, I would say it's not me but the bike. Some bikes will be like that.