The separation of church and state is not about preventing any kind of relationship between religious institutions and the government. Rather, it is about ensuring that the government does not establish an official state religion, and that it remains neutral towards different religious beliefs and practices.
This - but one could argue that the state having to make determinations of what constitutes a religious belief and practice _could_ go against the separation of church and state.
The state's role in determining whether a religious organization qualifies for tax-exempt status is not to evaluate the content or validity of the organization's religious beliefs or practices. The state's role is to ensure that the organization meets certain objective criteria for tax-exempt status, a necessary function.
You sure can if you follow the IRS's strict rules and requirements for granting tax-exempt status to religious organizations. I suspect you would have difficulty in demonstrating that the organization has a legitimate religious purpose and is organized and operated exclusively for that purpose. Creating a fake religion is likely to be viewed as fraud and could result in legal consequences.
