A]
Well, if it doesn't come out, it is either mounted on a pickguard or on pickup-rings. Either way you gonna have to remove the guard/ring before you can take the pickup out.

B]
Even so, a pickup has a wire coming out of it, so first thing is to disconnect that wire from the guitar's controls.

C]
You probably need to remove the strings too.

D]
Why take out a pickup unless you have a replacement nearby? (Silly question, I too like to peek into how things are made.) Be gentle with it. Don't rip any wires off (instead: un-solder them) and don't take the pickup apart.

the 2 screws simply detach the pickup from the pickup rings. you need to remove the rings as well. from there open up the electronics cavity and make a quick schematic of exactly what wires connect to where. from there you are safe to go ahead and detach JUST those wires using a soldering iron to heat up the connections and remove the wires. then you are ready to put the new pickup in. now most new pickups WILL come with their own schematics or you can also go online and find many sites that also carry schematics. seymour duncan has an excellent customer service link for schematics of thier pickups.

you will want to read some of those schematics to familiarize yourself with where the wires go. many cheaper pickups may only have two or three wires total. when you get a newer,better pickup ,you will notice 5 or so wires. two of them are usually not even used(simply wired together and off to the side). so don't freak out when you see more than you need. it's all good.

look at the website schematics to determine which setup you need to use(ie:single,single,single,,,humbucker,humbucker.....humbucker,single, single....etc.). there will be drop down menus to select which setup you will need. there will also be pictures of with toggle switch setup you are using as well..

gennerally speaking,changing pickups is a relatively easy procedure once you have read up on what u need to do. soldering is NOT difficult. heat up the area you are soldering to. put a little solder on tip of the gun(or simply use what is allready on the connection). apply the bare wire to the spot. make sure the wire has some solder on it and then let go and let it cool for a minute. double check your connection to assure it won't come off and you're done...

once you have wires connected, then you will have to re-attach the pickup to the actual pickup rings. this can be a little tricky getting the spring to stay in the assembly as you screw it back to the ring. i suggest taking the whole thing out and away from the pickup cavitly so you can get to it easily
removing the strings may or may not be necessary depending on how much wire you have to work with. at the least you will have to loosen up your string to slack to give you some more room to work with.