Hi, I'm going to answer this in parts.
This is not normal, not at all.
Is your screen brightness set to 100%? Are you running tons of apps in the background?
If not, it could be possible that your battery is faulty and is dying.
There are quite a few & honestly, I can only give you my own opinion. There is no standard "best ROM".
Try some of them and see what suits your device the best.
I personally love the stock ROM.
Rooting - yes.
Unlocking the bootloader - depends. Why do you wanna do this? Are you planning to flash a ROM that needs a custom kernel to run?
Are you planning on using a custom kernel or try and build one? If the answer to any of these is yes, then go ahead and unlock the bootloader.
Yes, rooting doesn't erase the DRM keys. You don't need to unlock the bootloader to root.
Check
post #2 in this thread. There's links there.
If you backed up your TA partition, you can restore it at a later time when you wanna switch back to the stock kernel.
There's a link in
post #2 for this.
Try it yourself and find out. Each ROM will have it's own list of bugs.
Again, this depends on how you see the Bravia Engine. I just think it's a sort of filter that increases color saturation and sharpness.
I honestly don't see much of a difference, yet I prefer to keep BE off.
Varies across ROMs. Try it yourself.
Well, avoid cross posting. I hope this answers most, if not all your questions.