First, you should note, most regular dvd players only recognize video dvd format well which should include VOB, IFO and BUP files, so if you want to play the burned dvd on any dvd players, you have to burn a standard video dvd, for your problems:
1. No, all standard dvd burner drive and writable dvd disc are right, such as both single layer 4.7gb writable dvd disc and double layer 8.4gb writable dvd disc are right for video dvd burner.
2. Yes, you need an extra good software to help you to convert the avi files to video dvd format then write/burn onto a dvd disc. RoxioCreator isn't good at this work.
3. All standard writable dvd discs are right, such as, dvd-r, dvd+r, dvd-rw, dvd+rw, dvd-r dl, dvd+r dl, etc.
4. You should convert avi or mp4 to video dvd format which should include VOB, IFO and BUP files.
If you don't know how to do that easily, you can follow me to use RZ DVD Creator, it can convert any formats video files to video dvd format and burn to any dvd disc(DVD+R, DVD-R, DVD-RW, DVD+RW, DVD-R DL, DVD+R DL,etc), such as convert any avi or mp4 files to video dvd format and burn onto any DVD disc, easy to use, insert dvd disc into your laptop dvd burn drive, run RZ DVD Creator, select the burner drive as the target folder, drag and drop the avi or mp4 files into the software, then press the START button, you will get a right video dvd, and you can play the burned video DVD well on your dvd players and xbox, you can yahoo or google search RZ DVD Creator and download it, easy to use, hope these infos can help you.