Can she not be killed while she's sober!?

No. She and Carol are seeking to undeify themselves. Apparently, large quantities of wine and/or meths consumed over extended periods can eventually achieve this result.

Keep it up, girls. You can do it.

Maybe she's a Cullen!

Charlotte must be really thick believing Carol was evicted. Anybody who's ever watched BB would know evictions are only done by public vote.*

*Jade Goody excepted.

Judging by the housemates reactions when they entered the house I doubt any of them has watched BB before!

Alcohol is her kryptonite. While sober she is immortal but after a few shots she is "mortal". Then she can be killed.

Unfortunately this is the only explanation.

The expression is 'mortal drunk' shortened to "mortal".
Not much different to "dead drunk".