I have a Schecter Hellraiser. I usually play with Hybrid Slinky's so that I can bend easily and all that whatnot. Well, I'm in a heavy band and decided to switch to a heavier gauge (Slinky's Skinny Top/Heavy Bottom). It took me ages to finally tune it how I wanted it. I love the deep notes of the low strings. The only problem is now it's very hard to bend the strings (any of them). Also, I can no longer fingertap like I was able to before. When I looked at the strings from a side angle, the strings were coming up way high toward the bridge/tremolo. 

Is there a way to fix this so I can bend strings and finger tap again? Or do I just need to get used to them?

Do I need to adjust my trust rod? If so, how? (I've never done it). 

Lastly, would it be bad to have the heavy gauge strings for just the low strings (EAD) and maybe the hybrids for the higher strings (GBE)? Or do they all have to be in the same family?

I play Drop A and B by the way. 
First off -- I don't know how you're even playing anything with those damp spaghetti noodles in tunings as low as Drop A and B.  I use EB "Not Even Slinkys" (.12-.56) and they're barely thick enough for Drop B let alone A.   

Anyhow, this is basic physics.  If you have a guitar setup perfectly for a certain tuning with a certain set of strings, and then you go to a thicker set of strings but the same tuning, there is going to be more tension than before.  When there is more tension between the bridge and the nut, it is going to increase the amount of neck bow you have, thus raising your action.  You'll have to compensate by lowering your action, but maybe adjust your truss rod as a last resort.  Generally, just lowering your bridge a little should bring you back to the action that was comfortable before.