What is the Bazaar and how is it different from the Auction House?
The Bazaar is your own personal store. In it, you can sell just about anything in your personal inventory (what you carry around, not items back at your Mog House). When you have a bazaar, a special icon appears next to your name and people can browse your bazaar and possibly buy your stuff.
The Bazaar is different from the AH in that there is no bidding. The price you set is the price it will go for.
How do I determine the cost of something?
Bear in mind that every sellable item in the game is a commodity. This means that if there is a glut in the market, the value of an item will be very low. Conversely, if there is a dearth (shortage) in the market, and yet there is a demand for it, the value of an item will be much higher.
Another thing to bear in mind is how needed/desired an item is in a given area or time. For example, antidotes may be worth much more in an area full of poisonning enemies than they are in the middle of a town where they can be purchased at local merchants. Or eggs with letters on them may be worth much more during the Easter festival than at other times of the year.
One final thing to keep in mind is that Bazaar items generally go for less than they would at the AH or in stores. So, if you are trying to sell something much higher than it can be found elsewhere, then you probably wont sell it (unless the demand is very high).
These things being said, check out the item's price at the AH, local stores, and see what you'd get to sell them to merchants. Take a price somewhere in the middle of all these things and use it.
How do I hide my personal info in my Bazaar?
The same way you do in-game. Go into "Config" from the main menu and hide your personal details.
Do I want to hide my personal info?
I don't know. Do you?
Sometimes it's a good idea (if you dont want any hassles like "Dude! Don't be a Elvaan WHM!" or "Woah! You're a 67 Paladin!") other times it's just being rude or elitist. Use your better judgement.