I'm new here, so hi.

It's time once again to change the strings on my guitars and I was hoping for some advice. I have a two guitars; a Gibson LP and a Firebird, and I use .10s on both of them. I keep my LP in standard but, I've been recently putting my Firebird in C tuning (not drop c). I notice that my Firebird feels a little too floppy, so I was thinking of putting .12s on it.

Is this a good idea? Is there anything I need to do before I switch to a higher gauge(adjust nut, etc..)? I plan on keeping it as my low tuned guitar, so I wont be straying from C any more than a half step.

Also, I sometimes drop my low E string to an A on my Les Paul. Should I not do that if I'm only using .10s?

P.S I plan on switching to Elixir Strings. Is there anything I should know about them before I switch?
yeah you should change the gauge if you plan on playing in C.
a set up MIGHT be necessary but see how it plays first and how everything is
(neck,frett buzz, intonation etc.) if everything seems fine then i see no reason for it.
Assuming you have TOM bridges which im sure you do.
I think you'll have to adjust your intonation when you go up to thicker strings, but I could be wrong. I use Elixir myself, damn fine strings. They cost a bit more, but they really do last longer, I use then 10-52's and after trying Elixir for the first time I've never looked back.