Switching from classical guitar to electric is going to take some time and practice. While you may be familiar with playing a nylon-stringed guitar, not all of those skills will transfer over easily or quickly to an electric. The best advice I can give you is simple: give it time, and keep practicing. It's not quite the same as learning the guitar from scratch, but you will have to adapt your playing style to accommodate the differences in string spacing and picking styles.
As far as some of the specific concerns you mentioned, the best thing to do would be to talk to an experienced electric guitar player or a teacher who can give you some feedback on your technique and what you might be doing wrong. I can give you some general advice, but playing with someone in person will go a lot farther and prevent you from forming bad habits.
The string spacing is just something that takes time to get used to. You will continue to miss the strings sometimes and pick the wrong ones until you get the muscle memory down that is required when playing electric guitar with a pick. It might help if you play fingerstyle some of the time on your electric, that way you can use the familiarity of something you do know, and focus on adapting to the different string spacing rather than learning a new picking method at the same time.
When using a pick, you shouldn't be touching the strings with your right hand thumb at all really, except when muting. In between chords or in quiet sections you can rest your thumb or the side of your palm against the strings to mute them, but when you are actually picking your thumb shouldn't really come in contact with the strings. Sometimes when picking higher strings, you can rest your thumb (specifically the fleshy part of where your thumb meets the palm) on the lower strings to mute them intentionally. On an amplified electric, you need to be more aware of strings ringing when you don't want them to, so when playing notes on higher strings you usually adjust your technique to mute the lower strings with either your right hand like I mentioned or with the extra fingers on your left hand. This is how a lot of guitarists can get away with just strumming all the strings all the time, and only picking out the notes they want to play and muting the other strings with left or right hand technique. Of course if you don't want to mute the strings, you just need to adjust your hand position and how you are holding the pick to make sure you don't touch the strings that are ringing.
As far as resting your picking hand on the body, I'm pretty sure that will also get more comfortable over time. I like Strat style guitars in particular because they are very comfortable to rest the forearm on while playing; some other electric guitar body-types aren't quite as accommodating. That is mostly just personal preference, but it will also get easier for you with more practice. Again, you are learning a new instrument- it's not going to be 100% comfortable right off the bat. This is another area where playing with someone more experienced that you will be helpful, as they can tell you if you are doing something wrong or if it's just uncomfortable because you aren't used to it yet.
I believe it is possible to play both electric and classical guitar well. It will just take practice, like anything else. If you want to stay good at both, obviously you shouldn't neglect your classical playing while you are learning electric, and be sure to continue practicing both. Metal guitar especially borrows a lot from classical styles in technique and playing style, and there are some great guitarists out there from all genres that have made the very same transition.