[Q] SGH-T999L ROM Info

This paragraph is intended to provoke thought and is not necessary for answering questions about ROMs, so skip past it if you want: When it comes to cell phones, I like freedom and options. Google has been criticized by the Electronic Frontier Foundation for censoring AdBlock from Google Play. The EFF also levied a criticism for Google doing away with AppOpps. I remember separately having a YouTube account and Gmail account, gracefully using both. After Google bought YouTube, the two were merged without any choice to keep them separate. Then Google+ was introduced and using YouTube became annoying without accepting a Google+ account. Eventually they shoved it down the throats of YouTube users and leaving a YouTube comment was blocked without creating a Google+ account. When you sign up for a new Google account, your searches are saved by default in the history unless you manually turn the feature off. Of course, the NSA leaks raise more questions. Google offers many efficient, valuable, convenient services but I want to maximize my freedom.

Some facts about my phone situation: I own the LTE-capable Samsung Galaxy S III SGH-T999L. It is rooted. It is locked to T-Mobile. Titanium Backup Pro, Root Explorer, and Terminal Emulator are installed. I backed up Knox with Titanium Backup and (as far as I can tell) uninstalled it. To my knowledge, there is no backup whatsoever of my operating system in its current state or original state. I do not know how to back everything up before I try flashing a ROM. As much as I want a non-Google ROM, I am unwilling to sacrifice LTE.

I never flashed a ROM before. Where do I begin if I want to do this safely and correctly? Is there a safe, stable, non-Google ROM tested to work specifically with the SGH-T999L and fully maintain LTE capabilities? Will I have to sacrifice anything major if I flash a ROM? Can flashing a ROM significantly affect battery life? Are there any important considerations you think I should know about before trying to flash a ROM (besides how to do it properly)?

Thanks.

I did not know that all Android operating systems were by Google. Thanks for letting me know.

When it comes to corporate tracking (e.g. collecting usage statistics in the background) or restricting phone options, is there any advantage that a custom ROM has over a stock ROM without bloatware?