Welcome to the world of backups.
Now, you can make your backups with your 360's drive, provided that it is a samsung model, hitachi drives don't rip the security sectors correctly, so you backups won't be good. Also, if using a 360 drive to rip, the drive needs to be flashed with xtreme firmware, ixtreme won't work to make backups.
The easiest way to make backups would be to obtain a kreon drive, check the stickies to know the model number, I always forget. If you are using a kreon drive, backing up a game is as simple as putting the disc on your PC and pressing start on XBC.
Last option would be using wxripper, but you'd need to get the SS and the rest elsewhere and that won't be discussed here.
And your burner should work, I use a similar LG model and works perfectly for burning, no coasters so far.
Always remember to make your backups on good quality media, singapore made verbatim is the best and the only you should use.
So, best option, get a kreon drive and you're set to backup your entire collection with ease.
