How is second income (full-time + freelance) taxed in Germany?

Hello ☀️ I am looking for some advice with some tax/finances questions. I am a software engineer, employed full time in Germany. I’m from the UK (EU citizen with no visa concerns, for now…). I earn >€56k, so I am already in the 42% tax bracket. I have only one income source – for my 2019 tax declaration I just had to enter the details on my Lohnsteuerbescheinigung.

I started doing some freelance development work for a client, charging an hourly rate. I don’t know what the end cost will be but let’s say for simplicity that it’s €1000.

My question is: how do I need to charge the client, and how much of this sum will I keep after tax?

I assume:

• I have to add VAT (Umsatzsteuer, 16%) to the bill, bringing it to €1160;
• When I receive the funds, I put away this €160 to another account for paying taxes;
• €1000 gets added to my total earnings, meaning it will be taxed at 42%, meaning I will keep €580 for myself;
• I put the 42% of the €1000 (€420) into another account ready for my next tax declaration;
• In summary, there is €580 for me and €580 (€420 + €160) for the German government

Questions:

• Is this assumption correct? If not, what have I missed?
• Do you have any advice for my situation? I’m not asking for tax evasion advice, but perhaps registering a business or renting a business might be more optimal.

I heard there is a tax-free limit on freelance income but I don’t believe this applies to my situation, being full-time employed. I will go to the Finanzamt in any case to get my Steuernummer and Umsatzsteuernummer, but I really like to go in with knowledge beforehand.