Tutoring on a fee basis. Taxes?

What about taxes here?

Of course, these must be paid if any arise.

This is how it is calculated

• Income from employment = gross wages – advertising expenses
• Income from business = Sales – Operating Expenses
• Income from self-employment = Sales – Business Expenses
• Income from renting and leasing = rental income – advertising expenses

Income from self-employment + any other income = sum of income and presumably also = total amount of income

Total income – special expenses – extraordinary expenses = taxable income

The taxable income forms the basis for your income tax burden, which is calculated according to the tariff in accordance with Section 32a of the Income Tax Act .

Any wage tax withheld by the employer will of course be credited against the income tax liability

I've heard that this counts as a trainer's allowance and that I can earn an additional €3,000 per year or €250 per month tax-free.

If the conditions are met, this is deductible.
Requirements are:

1. part-time
2. educational, artistic or caring
3. public or non-profit corporation

I think the first two points are uncritical.
Point 3 could therefore be critical and must be proven in case of doubt.

Do I still have to report anything to the tax office?

You must register your self-employment with the tax office. This is done via a tax registration questionnaire (FsE), which you must complete and submit electronically to the tax office. The tax office will then review your entries and assign you a tax number.