But is a lever an adjustable inclined plane, or is it a specific application of a wheel and axel?

And incline planes and axles are both types of levers 🤯

The difference between wedge and inclined plane is that the wedge uses friction to create a ratchet. You hit wedges but push stuff on an inclined plane.
The screw is similar, it can be screwed continuously or ratcheted, and it will rest in its final position under the same friction. Though you can also see stuff like locktight lock it in place when the friction is insufficient, so in theory you could have a screw that self unscrews without that.

The pulley transfers the force/energy through it from the ropes end, For the axle, one end is combined with the axle and rotates with it. This I would say is a special case of the pulley, but also is quite bad. You need to slap a rotating lever on the end as a big handle to actually gain anything.

The pulley nor the spool give you mechanical advantage. For pulley systems the advantage comes from combining the redirection properties of pulleys, and for spools they come from the handle acting as a lever.
The spool doesn't really redirect thogh, you could tie the rope directly to the handle and get the same advantage. The spool is just to collect the rope so you can keep turning without reattaching the lever. It extends your stroke.

Alright there Citizen Doctor Abraham Mehermblur

and aren't inclined planes just a fancy wedge

Screws convert rotational energy into directional energy