Here is what happens on each of the events you state above
I form a new LLC today
No tax consequences. You merely have to pay the filing fee associated with forming the LLC.
I infuse the company with $500
You may owe tax unless you do a certain transaction in which you take back a membership interest with the same basis as the cash you put in. This is what most founders do without even realizing it.
I do some work, and invoice a client for $250
No tax yet, but since it is a "pass through" entity, this automatically gets attributed to you personally regardless of whether or not you take the money out of your bank account. Therefore, you must pay personal income tax on it at the end of the year.
I write a check from the company account for each of 2 partners for $100 each.
If the owners also own a percentage of the LLC (i.e. they are LLC members) then actual payment of the $100 is not necessary for them to be taxed on their pro-rata earnings.
For example, say there are three founders that each hold 33.3% each. The LLC makes $300 and the year closes. Even if the LLC keeps the money in its bank account, each of the founders will be taxed on the $100 as if they had taken the money out. This is the nature of how "pass through" taxation works for tax reporting. You pay tax on it regardless of whether or not you take the money out.
The year ends. In this scenario, how do the owners report the $100 income on their personal income taxes
The LLC will have to file a form 1065 with the IRS by April 15th of the next year, and fill out a "K-1" for each of the founders, which they will use to report on their personal income tax return, or 1040s.
Does the LLC provide a 1099-MISC?
No, they provide a K-1 instead.
You don't have to get into payroll taxes unless you select what's called a "manager-managed" LLC and select officers of the LLC like a CEO, CFO, etc... When you do this, then you have to file quarterly payroll tax returns form 941.
If you want more simplified tax reporting, select a member-managed LLC and you don't have to file form 941.
Note: This is not legal advice, and does not create an attorney-client relationship.