1) Busybox couldn't possibly have caused battery drain. Almost every person who has rooted their phone uses it. I'd recommend you reinstall it. You'll need it at some point down the road, I'm sure. If you have any battery drain, I'd recommend you use spare parts to see what is using your system resources.
2) TB is certainly not the only option for backup. Others have been stated in this thread, already. You'll want something that does a full backup, though, like TB. MyBackup Root is a good option, I've heard.
3) There are many, many, different ways of removing bloatware. To name a few:
a) Flash a custom ROM. This will save you the trouble of doing it yourself. You'll almost always get busybox, as a result, as well. Of course, you have to take in to consideration what you potentially gain and/or lose from doing this. It really depends on a lot. If you want to stick with stock, this isn't for you unless you want to flash a close-to-stock ROM with bloatware removed.
b) Remove bloatware from /system/app. This is not recommended, as you can very easily break things if you're not careful. Always make a nandroid backup before doing this. You'll need an explorer capable of running as root (Root Explorer) or, for a free option, use the terminal emulator. You need busybox, though. To do so, open the terminal, and type these commands:
su
mount -o remount,rw -t yaffs2 /dev/block/mtdblock4 /system
cd /system/app
rm com.whatever
A much better option, though is to:
c) Rename apps. Basically, do the same as the above, but instead of removing them with a file explorer (or using rm in the terminal), just change the apk extension to .bak. So, for example, from the terminal, instead of the rm command, run this:
mv com.whatever.apk com.whatever.bak
That will ensure that the app will not run, and it will appear as though it's gone.
d) If you want to do *exactly* what TB does (freeze bloatware) you can do this from the terminal emulator. I don't think you'd even need busybox. This is how you do it:
su
pm disable com.whatever
It's as simple as that, and it will keep the app from running or showing up in your app drawer. To "unfreeze" the app, just run:
su
pm enable com.whatever
This is as close to TB as you'll get, and it's free. Of course, it takes longer than doing it through TB, but it's ultimately the same (and, again, free
)
e) If you tell me what you want to remove, I can make a flashable zip for you that will do the "rename" method above from recovery. The problem with doing this, normally, is that everybody wants something different, and it would be a pain to maintain a bunch of versions.