So I am pretty sure it just qualifies as a hobby as long as I do that.
No, the IRS qould qualify it as a hobby if you are not running it as a serious business (among other criteria). If it is classified as a hobby, you would not be able to claim a loss from it (if your expenses are more than your revenue). Whether or not it's a "hobby" or "business" you still need to report your income to the IRS.
Since you are not incorporated, you actually cannot deduct your expenses as you are claiming. That is a benefit the IRS only gives to businesses.
Compeltely wrong- you do NOT have to incorporate to claim deductions. You use Schedule C if you are not incorporated.
The game's user base is pretty international.
No relevence- as a U.S. citizen you are taxed on all your income reguardless of the source or location of the source. If you have bank accounts in other countries that people are paying into, you also have additional reporting obligations.
We make a small amount every month...
Who is the "we" to which you refer? Is it you and your spouse or you and one or more partners? If the latter, you DEFINITELY need to look into setting up something legit (whether a partnership, LLC, or incorporating), especially if you are planning to grow in the future. Not just for tax issues, but also legal issues to protect yourself in when a major disagreement arises.