Should I come up with a game company name?

Would it be fine to put it up under just my name?[/quote]
In many places this is fine (but see above) and it is a matter of choice. However, if you are making money, most locations require that you either register a company or at least register as self-employed with the tax authorities. There are differences in relation to tax and expenses between trading as an individual or trading as a company - you should talk to an accountant to find out what these are. If this is your first game the likelihood is that you wont make much/any money and so registering a company is an unnecessary expense.

Should I make a company name? (to make it look more professional)[/quote]
If you want this to be a long term venture then you can do.

How do I register the name so noone else can use it?[/quote]
A company name is considered a Trademark and in most countries you get automatic protection under trademark law when you start using a name/logo to trade. You get more/better protection if you register that trademark with your local Trademark office but, given you are just starting out it probably isn't worth the cost yet.

The most common being "If you made it it's already copyrighted by you", which I think is not true?
[/quote]
If "it" is a game then it is true. Copyright protection is automatic.

If "it" is a game then it is true. Copyright protection is automatic. [/quote]

So when I put my game on-line and someone steals the idea or game. I can tell them to stop, and if they don't I can sue them?
There's no need to go to a patent trademark or copyright office?

Just double checking.

The moment I put my game on-line I can put under it © coderwalker ?
Is something else required before I start putting © coderwalker ?

I am in NC US and no, no start-up capitol what-so-ever for it. It being a game.

Most people use company.com and company games. Am I allowed
to say Coby Walker Games or would that be a trademark I don't have
registered?

So when I put my game on-line and someone steals the idea or game. I can tell them to stop, and if they don't I can sue them?

There's no need to go to a patent trademark or copyright office?[/quote]
You don't have too but (in some countries) you gain extra protection by registering copyright or trademark. However this all costs money and if your game never makes any money it is pretty pointless to pay out for extra protection that will be worthless.

The moment I put my game on-line I can put under it © coderwalker ?[/quote]
Only if that is your real name or the name of a registered company. Only a legal entity can hold copyright

Most people use company.com and company games. Am I allowed
to say Coby Walker Games or would that be a trademark I don't have
registered?[/quote]
You are confusing domain names and trademarks. They aren't the same thing.

If "it" is a game then it is true. Copyright protection is automatic.

So when I put my game on-line and someone steals the idea or game. I can tell them to stop, and if they don't I can sue them?
There's no need to go to a patent trademark or copyright office?

Just double checking.

The moment I put my game on-line I can put under it © coderwalker ?
Is something else required before I start putting © coderwalker ?

I am in NC US and no, no start-up capitol what-so-ever for it. It being a game.

Most people use company.com and company games. Am I allowed
to say Coby Walker Games or would that be a trademark I don't have
registered?
[/quote]

Copyright is automatic, you don't even have to put a copyright notice anymore (Allthough in the US it used to be required and it is possible that it will help you defend your rights in some jurisdictions (ask a lawyer)), defending your copyright however is expensive, copyright does not cover ideas and ideas can not be stolen (You don't have exclusive ownership of your ideas unless you keep them to yourself, and even then other people have the right to come up with the same ideas (and most likely someone allready has come up with the very same idea as you and either didn't consider it to be good enough or didn't have the skill/resources required to pull it off)), it never hurts to make it clear who the copyrightholder is since works with an unclear copyright status is a pain in the ass for everyone.

You don't have to register a trademark, however registering your trademark will help you protect it, for an indie developer the costs involved in defending a trademark probably isn't worth it, registering might be worthwhile depending on its cost just to make your name listed in a commonly searched trademark database (alot of developers will search the us trademark database to avoid problems) but having your game reasonably well known will be protection enough against honest mistakes.

I don't know about the US but in sweden its perfectly fine to do business as an individual but taxation is a bit more problematic since any revenue will be taxed as income and you can't make tax deductions for private expenses (thus if you have $90.000 in expenses and get $100.000 in revenue you'd pay income tax on the $100.000 as an individual while as a company you'd only pay tax on the $10.000 profit) (It is possible to do software development without any significant expenses other than your own time though)

you don't even have to put a copyright notice anymore (Allthough in the US it used to be required and it is possible that it will help you defend your rights in some jurisdictions (ask a lawyer)),

Trademark law, however, is very very different and requires registration, validation and (possibly) a fee or another.

[quote name='DarklyDreaming' timestamp='1298463200' post='4777918']
Trademark law, however, is very very different and requires registration, validation and (possibly) a fee or another.