Allright I am a newbie such as you but I can answer a few of your questions.
3) After I work out, it seems that my muscles are bigger. But after I go take a shower, they slowly start to deflate to normal, and later it seems as if I had never worked out at all. What's happening?
As you are working your body the muscle is being used and more blood is being pumped into them making them seem bigger as you cool down the blood flow goes back to normal thus making your muscles look smaller.
4) After workouts, does it matter if I take a hot or cold water shower?
Hot is the way I go it helps to relax your muscles so your not as sore. My folks have a hot tub I use which when I first started lifting helped me abit.
5) How many days should I work out a week, and for how many hours?
I work all week giving sunday to breath BUT I don't work my whole body out. I rotate on a 3 day plan my old man uses.
Day 1 Chest/Arms
Day 2 Abs
Day 3 Legs
Day 4 Chest/Arms
Day 5 Abs
Day 6 Legs
Day 7 Rest
This way I give my body time to heal and rest up.
Worked very well with my old man so I also do it this way.
I also only work out for about 25-30 mins this is not including the time it takes to walk and run 2 miles.
6) Would jogging/swimming help build muscles? (Well this question is kind of common sense, but I mean it in a "would it be better to be jogging than lifting weights" way)
Working out with weights is far better then just jogging/swimming BUT I would do both of them if possible. I currently walk 2 miles with my 9 year old dog (for a warm up) and then run 2-3 miles every night.
7) Is it true that doing push-ups and only push-ups will eventually make you "big?"
If it does then I feel cheated. They do help but doing just them won't give the best results.
8) Should I buy more equipment? If so what would be most cost-effecient and beneficial?
When I was 14 I went to wallmart and got a cheap bench for 69.99.
It came the standard 10 pound bar with 2 Dumbells and 150 pounds of weight.
For the price I payed I got my moneys worth out of it. It is also in the budget of a 15 year old

This will at least get you started and if you end up hating working out...then your only in the hole 70 bucks. I would take a look at your local stores and post the link to what is in your budget im sure the people here can point ya to a good bench.