What type of tool is an “extension”

Choose high quality for the most commonly-used tools.

There are a lot of crappy tools out there, many of them sold by Harbor Freight at rock-bottom prices. Those tools only make sense if you plan to use them very rarely, and if you will use them very gently, and if you can't rent them nearby.

For the tools you use most often, buy the best quality you can find. Working with good tools is a joy; working with crappy tools is very frustrating.

Don't buy every tool you could imagine using. While "the right tool for the job" makes the job go smoothly, buying all the "right tools" means you'll spend all your time organizing your extensive tool collection.

Buy tools that match the work you like to do. I haven't seen your bed, so I'm not sure how it's put together. It sounds like you plan on only very minor tinkering. With that in mind, I recommend you buy:

• an adjustable wrench
• a flathead screwdriver
• a philips-head screwdriver

These 3 tools will let you do a lot of tasks. I'm guessing that will be enough for you. Beyond that, see the question that @NiallC linked to.

Regarding kits:

Kits sometimes sell on false abundance. That is, you see 100s of shiny bits and pieces and think "everything I could need, wow!". They may have duplicates (my socket set has four 10mm sockets!). They may have items you'll never use. And they may have low quality, hiding behind the size of the collection.

Kits often come in a nice case, making it easy to organize and store your tools. Unfortunately, if you need 1 more bit/socket/attachment/whatever than what comes in the kit, you don't have a good place to put it.

If you haven't already, the cross-hatched drain cover in the tub can be removed with this wrench, called a bath tub drain wrench (or similar):

There should only be plumbers putty underneath it, so it will be hard to turn at first, but then quickly becomes quite easy.

Once that is out of the way, running a small drain snake down the drain is comparatively easy. There is probably no need for you to go under the floor.