How do I know if the chainset is compatible with my bottom bracket cartridge?
Look at the type of the bottom bracket to crank attachment. Most likely it is square taper. Then look at the bottom bracket spindle length. For example if you have 110mm spindle yet the new crankset requires 130mm spindle, the chainline will be affected by half of the difference, i.e. one centimeter. That's a lot! Also it's possible that the cranks hit the chainstays of the frame so not all cranks fit with all spindle lengths. Also, there is a difference between JIS vs ISO square tapers. So not only does the length have to match, the standard of the square taper should also match. (It might be possible to use a JIS taper with a slightly differing length on ISO cranks or vice versa.)
Are all shimano cassettes compatible if the speed matches?
Based on the information, you do not have a cassette but rather a freewheel. The freewheel will be so tight you'll hurt yourself if trying to remove it, if the bicycle has been ridden as opposed to being only parked. And you can only replace it with a new freewheel rather than a cassette, unless you change the hub (which is the second most difficult part in a bicycle to change -- unless you replace the entire rear wheel).
Am I right that replacing a 14-34T cassette with a 11-28T cassette will result in less gear range on the higher and lower ends, with more similar gearing?
You'll have a new higher gear with 11-28T cassette but the lowest gear you used to have with 14-34T freewheel won't be available. Of course if changing the chainrings, you can compensate and besides 28 tooth chainring and 28 tooth sprocket is so ridiculously low gear it should be enough for most purposes.
Is it a bad idea to buy a chainset instead of individual chainrings? It seems like a cheap component and might be a simpler process?
There's a reason for being cheap.
The cheapest cranksets have rings made of steel. They are heavy and often times the steel used is also quite soft so not necessarily that durable. The slightly more expensive ones have aluminum chainrings but made of a soft alloy. They wear in no time. Only the very best chainrings are made of high-end aluminum alloy such as 7075T6. By purchasing a cheap crankset, you won't get the very best chainrings.
If your bicycle is a 2009 bicycle that has a freewheel and not a cassette it's probably a very cheap bike so not necessarily a good idea to throw a lot of money at it. So you might want to either use a cheap crankset or much better yet, reuse the old crankset. Old chainrings usually work with a new chain if they worked with the old chain too. Only for rear sprockets is the failure mode such that they refuse to work with a new chain. So you don't necessarily need to change the chainrings or the crankset.
Any other compatibility problems I've not spotted?
Yes. The freewheel vs cassette compatibility problem.