Chains wear out pretty fast, especially if they're dry or dirty. Worn chains cause cogs to wear much faster, in turn. Change your chain regularly before it's too worn and you can keep your cogs longer!

In general you can use any higher-speed chain on any lower-speed transmission, although people tend not to because higher-speed chains are higher-priced. They need to be equally strong but in a narrower space, and the machining is fairly precise to make sure the shifting works smoothly. That costs more.

Use whatever brand strikes your fancy unless you're using 10-speed or higher. Then it becomes little bit more finicky and less inter-compatible.

Quick-links are convenient and foolproof (for a given value of fool) and are the only reliable way to splice 9-speed and higher chains, but they are only reusable a couple of times. 11-speed and higher shouldn't be reused. Wipperman Connex links are the exception: they can be reused more or less forever. They're a good investment.

More expensive usually doesn't mean it lasts longer in chains. It usually means it shifts a little bit better or is a little bit lighter. That might or might not be important to you.

Track or fixie transmissions usually use wider 1/8" chains and won't work at all with the narrower multi-speed chains for derailers.