Hey, welcome to Amiga.org :-)
I'll try to answer some of your questions.
1. ADF stands for 'Amiga Disk Format', and ADF files are basically an image of an Amiga disk, uncompressed, so all ADF files are 880k. There are some legit ADF's out there, but the majority are illegal, ie uploaded without the copyright holder's permission. Even though the majority of them are not available for sale, they are still under copyright, and I don't want to spark a copyright morality flame war here, so I'll leave that as it is. :lol:
They are indeed transferrable to the Amiga, you can do it a few ways - connect a null-modem serial cable between the PC and Amiga and use a program like Transwarp (I believe that can do it) or Amiga Explorer. You could also use network transfer if your A4000 has a network card, or burn the adfs onto a cd and write them onto floppies from there. Another option is to 'install' them onto your PC using WinUAE and WHDLoad (a Hard Drive installer program for a couple of thousand games) and then burn the installed folders onto a CD and copy them directly onto your A4000's HD. (Sorry for the lengthy response :-D )
2. You can use an IDE CD-ROM (Any one should do, even new 52x burners) or a SCSI CD-ROM if you have a controller card installed. IDE is probably easier, but I'd imagine you'd get better results from SCSI (Less load on CPU). You can also run CDTV games on it (I believe, may need to use software to 'degrade' the processor.) and CD32 games with the use of software to 'emulate' it, as the A4000 does not have the Akiko chip that the CD32 did.
3. BTTR does indeed have downloadable games, and you can download them from your PC. I believe they're probably in LHA format, which needs you'll need to decompress them from the Amiga side.
Here is the ADF games section, use the alphabet at the top to browse.
4. PAL games should be playable on your NTSC Amiga, if you switch the output to a PAL screenmode. If you install the game onto your HD using WHDLoad, it should take care of that and run it in your existing screenmode (I think :lol:)
5. To run your Amiga 4000 on a newer monitor, you will either need a multisync one (one capable of syncing down to 15khz) or one that syncs to 31khz and a scandoubler for your A4000 (Does what it sounds like, doubles the sync rate). Some people have had good results with LCD monitors and LCD TV's.
Good Luck :-D