Moin Thomas, sorry you are having trouble getting your forms to work.
Lets run through some checks! i bet we can get this working.
I realize you said you had read some guides already, but this is the best guide to assist with the problems you are experiencing. did you read it? If yes, cool, i just want to check.
Forms absolutely do work on the starter tier, with up to 100 submissions per month.
One more sanity check: when you are testing your form, are you using data such as "test@test.de
or similar? because those will definitely get filtered out by the spam checker.
Also, I see that your name="Kontakt" etc attributes are uppercase. I am not 100% sure whether this makes a difference, but normally people lowercase everything. Can you try changing all the name attributes to lowercase just to be sure?
I might also suggest you simplify the form so we can get it working (for example, take out the spamcheck honeypot field - we can always put it back in once we figure out what is causing the problem.
Can you try these things and let us know? we definitely want to help you figure it out.