Type and strength of a typical chain I'm crafting a small 3 speed "gearbox" for a project of mine and I'm aiming for simplicity in the first place since I'll have to create it myself and I'm not particularly gifted when it comes to using my hands.

Initially the idea was to buy proper cogs but when I discovered how their prices go I resorted to using bicycle sprockets from a cassette like this one which go for ~$10. 

Basically I'll take it apart and have 2 shafts with sprockets on both sides matching my calculated gear ratios, one side will be on bearings, all 3 pairs, respectively for my 3 gears, connected with a chain piece of their own and I'll have selectors sliding on the bearing side engaging different gears - pretty close to how a conventional gearbox works just a poor man's version.

My problem is I can't figure out what type of chain goes on a typical mountain bike you would see sold commonly in any casual bike store. There are a bunch of different claims I found which lead me to believe that it's either 25H or T8F but I couldn't determine with enough confidence to order.

The maximum force that will need to be exerted in order to propel my vehicle on the steepest hill I've designed it for is about 650N, which multiplied by wheel size comes out to about 150Nm on the output shaft. That's roughly about 60kg I'm assuming the chain and sprockets will have no problem handling that.
There are several different sizes of chain based on the number of sprockets used in the rear cluster (e.g. 8-speed, 11-speed).  These are 3/32" nominal width.  Single sprockets for these chains are available, but can be hard to find and/or expensive.  In your application you may be able to get away with a slight mismatch, i.e. using a wider chain than your sprockets would suggest (an 8-speed chain on 10-speed sprockets, for example).  This wouldn't be a good idea on an actual bike, and it could get fiddly if you mix and match too much.  Here are the relevant spacings.

The alternative is singlespeed components - they use 1/8" chain and sprockets are always sold singly.

If you're going to pick up cheap used cassettes, I suggest you pick one type and stick to it.  All 8-speeds should work well together, or all 7-speeds, but less so for higher counts.  7- and 8-speed parts are cheap if you want to buy some new, while 5- and 6-speed can be hard to find and 10-speed or higher will get expensive.

As for the strength, consider the forces involved in hard pedalling - let's say a 100kg rider standing on a 175mm crank with the chain on a 22-tooth chainring.