Inexpensive 1x10 gearing? Can I use a single speed crank (like a Shimano Vee) paired with an 7 speed rear derailleur and 10 speed 12-32 rear cassette to make an inexpensive 1x10 gearing, for non-technical trail riding?

Is there something special about the more expensive components that are devoted to 1x gearing?
Dedicated 1x systems do have some specific features:


11 or 12 speeds
Derailleur able to handle a super ratio cassette (at least 11-40)
Narrow-wide chainring
Clutch system on the
derailleur.


You can definitely cobble together parts to make a 1x10 (or 1x9) system. A Vee crank will most likely be fine. 12-32 cassette is reasonable (11-32 would be OK too). A 7 speed derailleur won't work. 

You'll want to consider your overall gear ratio spread vs size of the ratio steps. Trying to run a cassette with a too-wide range means you'll have big jumps between gears, and requires a derailleur that than handle the largest sprocket size. If you settle for a narrower range cassette (sacrificing either high or low ratios, depending on what size chainring you select) gives you more derailleur choices. A narrower range cassette also means the derailleur has to deal with less slack chain when on the small sprockets and hence less chain slap.

The rear derailleur does not care what crank you use, or the chainring size. Just select a crank that fits your bike, gets the correct chain-line and has a bolt pattern and diameter that will take a chainring of the size you want. Narrow-wide chainrings are a good idea for better chain retention.

You need a derailleur compatible with a 10 speed shifter. For Shimano 1x10 you will have to use a 10 speed derailleur because Shimano 10 and 11 speed MTB groups use a different derailleur actuation ratio than 9, 8 and 7 speed groups.  The derailleur also needs to be able to handle the largest cassette sprocket