Freehub and spacers
 

I purchased an aluminum road bike that came equipped with a 7-speed cassette. I am looking to upgrade the wheelset to carbon fiber and go with either an 8 or 9 speed Shimano cassette and, of course, make the necessary gear shift upgrade. The Superteam set of wheels I am looking at says the freehub will accommodate Shimano 8, 9, 10 or 11 speed cassettes. If I purchase either an 8 or 9 speed cassette, will I need spacers behind the cassette or is that necessary only if I go with a 10-speed cassette which I understand is narrower than an 8/9?

I do not think you require spacers for 8 or 9 speed cassettes.
Dime a dozen, however, and you will find out when screwing on the lockring.
You only want to see the cassette firmly attached without any jiggling.
Perhaps your biggest challenge could be finding a cassette that will slide on properly.

No issue with 8 or 9 speed cassettes. Generally speaking, also no issue with 10 speed cassettes. Some 10’s require a modest spacer behind them, easily located at your LBS, Amazon or otherwise online.

Does the DO spacing (130mm?) match the new hub DO spacing? Same axle diameter & attachment method?

I had a bike with quick release 7 speed dropout spacing (126mm). Without opening up the rear triangle to 130mm the only way to get an 8 speed drivetrain was to buy a special 126mm hub with a 9 speed freehub body and use 8 of the 9 speed HG cogs. I don't think Shimano makes that rear hub anymore. Then I needed to build the wheel. You will want to be cautious cold setting an aluminum rear triangle.
See here: https://sheldonbrown.com/shimano-cassette-bodies.html

Since the freehub body can accommodate an 11 speed cassette, you would need to add a 1.85mm spacer behind the cassette if you use an 8, 9, or 10 speed cassette. There is a chance the wheels might come with one, otherwise they are quite cheap to obtain.

The DO dimension at the rear is 135mm, with 100 at the front. For whatever reason, even though the manufacturer chose open dropouts front and rear, the bike came with a QR skewer at the front but a 9mm axle at the rear with hex nuts on both ends to hold it in place. The new wheelset comes with QR skewers front and rear.

That's unusual, a road bike with QR and older MTB dropout spacing. Anyway, you're good to go.

I might save that money towards a more practical bike for your usage. 7 speed is pretty old stuff so anything newer is not going to be of much quality so putting a ton of money towards it to upgrade with carbon wheels and a new groupset is a bit much. If you are dying to spend that money it is in the end your money but I would save it towards a bike with parts you are more interested in.

You seem confused, 8 sprockets of 9 fit on a regular 7 speed hg freehub with 126 mm dropout spacing, no need for a special hub.

The lone qr is the mark of a bike that is built down to a price. Nowadays 7 speed gears also indicate cost cutting. I wonder if carbon wheels are appropriate for such a bike - does the bike even have disc brakes, or the fittings to mount them?

The bike does have disc brakes