Hey there!  I'll try to answer most of your questions!
1. What are the necessary items needed for owning a horse? (Tack, grooming, etc.)
You'll need a few good brushes, curry comb, hard brush, dandy brush and a brush for mane and tail, would all be great for basic grooming!  You also will
 NEED a hoof pick, to pick out your horse's hooves before and after riding.
You will NEED a good halter and lead rope!!
The other kind of tack you will need depends...do you ride English or western?
For western riding you will need:  Obviously, a western saddle, western saddle pad/blanket, a good cinch, (NOT A GURTH.)  You will also need, a bridle, a bit, (snaffle bits are the most common bit used, and best for riding in, Imo.)  Or if you decide to go bitless, you can get a bitless bridle or a hackamore.  A breast collar is good to have also.  I keeps the saddle from sliding back if you're going up hills etc.  Some people don't use them but I usually do almost every time I ride.  I feel like it keeps the saddle more secure..
If you're doing English, you will need: an english saddle, an english saddle pad, a gurth (NOT A CINCH), a martingale, which is optional.
You'll also need an english bridle, whether you use a bit or get a bitless bridle is up to you.
2. How often should you ride your horse?
How often could you ride?  The more riding the better for you and your horse.  5 times a week is good in my opinion, but everyday is even better!  To exercise your horse, you don't have to just ride...lunging up hills and over small jumps etc is all good for your horse to get exercise, build muscle, and is also good for training etc.
3. What are good companion animals for horses?
Another horse would be best..  Some horses get along well with a goat or a cow but other horses are can be very cowy and aggressive towards them.
A miniature horse or small pony usually works well also.
4. What ingredients should I look for and avoid in a horse feed?
Not really sure.  My horses don't need horse feed very often, they stay stout on hay and grass.  Sorry about that...
5. I'm sure horses can't only eat hay and graze, but how much of their diet can just be those?
Actually, horses mainly do just eat grass and hay..  Most people out here in Oklahoma have large pasture fields and don't need to give anything else besides hay in the winter and sometimes pellets..
6. What's a good horse age for a beginner to get a horse?
That depends on the horse's training and temperament... Usually older horses give lessons to young kids and inexperienced riders and have a nice calm personality... 7 to 18 would probably be ideal....but I have a  2-3 year old, 12 and a half hand pony, that is the best kid/beginner horse ever!  3 year old kids can ride her alone without help and she's amazing with them and has such a calm personality!  Then I've had older horses that are full of sass and like to buck, play, etc...So it basically all depends on the horse.  I wouldn't pass up a good horse just because it's young..
7. How often should you groom a horse?
Everyday or every other day is good, but you
 NEED to groom your horse always before and after you ride.
8. How often should you muck out the stall?
It should be cleaned ever day if your horse is kept in it a lot.
9. How often should you change their shoes?
4-6 weeks.
10. Do they really need shoes?
No, most horses don't need shoes, but some horses do have sensitize feet that crack without them...  All of my horses do just fine barefoot.
11. What are some good beginners horse keeping books and/or websites?
You can find a lot of good horse books at the library.  They should have everything from basic horse grooming and care to more advanced riding and training info.
12. What's the best bedding for a horse?
Straw works really well!
13. Is it cheaper to board your horse or to keep it at your home (if your able)?
Keeping a horse on your own land would be a lot cheaper I believe, especially if you have a good grazing pasture.