The more scared a person is, the more energy is given off. That is the energy that makes dogs bark at you. That is also what attracts entities-whether it is a ghost or an entity of some sort.
I will answer each question in the order it was listed:
1. Learn to put up shields-but I know, it's hard sometimes. Right now I am trying to ignore the whatever that lives in my closet. It sounds crazy, but I have pictures of it.. I'm not sure what it is. But just mentally say to yourself that you don't want to be watched and to be left alone from anything negative.
2. Always check to see what it was. It may have been a car moving by the house or some animal going past the window, etc.
3. If you can get a camera with a camcorder in it, or just a camcorder, set it up at the door where it won't be disturbed and leave it for many hours. This is what ghost hunters do to catch activity on tape. You may have to go through long hours of taping. But if this is possible, this will help disprove that this is the wind or someone else moving the door.
If you can't set up a camera, just try to get others to watch out for it. If there is a witness then you can prove that the door moved on it's own. One entity that bothered me for a few months made its presence known at my work. It was drawn to my coworker who was sensitive to that sort of thing, so it would make cartoon noises or try to pop off a top to a hairspray. It wanted attention.
4. Again, take pictures of whatever your cat is staring at. It can show up on film. Hopefully you have a digital camera so you can see something right away. Sometimes you have to search for it... Even then remember that it could be 'anything'. I used to think my cat wasn't that bright because she did the same thing. But she proved me wrong...
5. This can go either way-either you need a plumber, or something wants attention. I wouldn't read too much into it, though.
6. It sounds like there is a lot of energy... whether it's coming from an entity or you, I can't tell from just that statement. Something is most definitely there though. Maybe you are just getting freaked out and the energy is messing up electronics. Get an amethyst and put it on the electronic. It should straighten it out. I knew a girl who used to just walk by a computer and it would turn itself off until she started to wear an amethyst. She had that much energy. Learning meditation to find balance would be good for you.
7. Consider the surroundings-if you are in humid weather and you get goosebumps, it could be something. Take pictures to find out.
8. This is where it will sound weird, but these actions works wonders! Do not store ANYTHING underneath your bed. If you have any religious/new age material, store it in the highest place in your room. For example, if you have a Bible or tarot cards, but it up on the top shelf in your room or closet. Never store that under your bed because when you sleep, you are trying to go into a 'higher' or state of consciousness. Any material meant to bring you 'higher' should not be 'lower' than where you sleep. Remember what is usually thought to be lower.. Hell.
The weirdest but best advice I got when I became aware of the activity was to put sea salt around my bed. Put it so that there are NO gaps, around my bed so that the area is encircled. I think you are supposed to crush the salt into the carpet, but then it's unsure to me where the gaps could be. I once had a gap in the circle and I had a dream similar to yours. I felt frozen and I saw something trying to crawl to me. When I became aware of it, I fought to wake up and I did. I fixed the gap and it was fine again. If your parents ask you what you are doing (and they will) just tell them that you have been having bad dreams and you heard that this works.
I think the entity was trying to get your energy. You may have a lot because you are psychic or maybe it's just because you have a lot of emotions going on (part of being a teenager, it calms down after your mid 20's at the latest).
I had an incubus problem (look up the term incubus if you want to disturb yourself). I don't have a problem anymore, but that is because I still have salt around my bed. It's silly but it works.
Look into other crystals for protection. Say a prayer over the one you choose (or charge it if you are Wiccan, etc) and keep it with you.
Definitely memorize the St. Michael's prayer listed.
Whenever you don't know what to do about a paranormal problem, research it on the internet. It's good that you found this site, and I hope it helps. Please always try to find logical explanations to something. If you think it's something else, you can freak yourself out. So prove that there is no logical explanation for it-remember when you are tired everything seems strange..
This situation can stress you out, and it sounds like that is what this entity wants you to do. Stay clear of drugs, etc. Make sure not to eat any weird foods before bed (read:junk). I heard chocolate and other junk can cause nightmares. No caffeine before bed either.
If this doesn't work, write back on this board and I will try to help. Or at least steer you in the right direction.