World of Warcraft: Can I mail gold from my horde character to my alliance character on the same server?

Not directly. However, you can utilize the neutral auction house in booty bay. On your alliance character, put an item up for auction and set the price for the amount of gold you need + the tax that the auction house takes. Then buy the item on your horde character at the neutral auction house, or have a friend buy it. Your alliance character will then get the gold in the mail.

Alliance To Horde

1

Horde Characters

i think it can also share you a coupon site
to got gold quickly
http://www.coupons.com/coupon-codes/igxe/

"Mail can not be sent to characters in the opposing faction (i.e. Alliance characters can't send mail to Horde characters.) "
http://www.wowwiki.com/Mail#How_do_I_send_mail.3F

Exists on some privates but not legit retail.

I haven't played it for a few years, but I believe no, you can't. There are no interactions between the two factions like that.

You can technically swap items via the Gadgetzan auction house, but it'll cost you.