After failing 3rd time you will not get slot soon?

The first 2 tests, you get accses to some more times not everyone sees. Everyone thus gets a chance to do it.

If you want a test really soon, look for canceled tests. Somtimes someone will cancel last minute and you can then easly pick up a time on short notice.

For example: On one testing center i checked you would have to wait until June 4th to get a driving test - but checking a few times in a day I could find one on May 20th and even April 29th. You just have to look constantly and book if something comes up.

I know there is priority for the first two attempts, but after failing the second test yesterday, now I can only book test in exact three months and at that time my theory results will be expired, am I blocked by Trafikverket to book a closer date for the test?

You are not blocked at all. If you find an available time you can book it.

Go onto the website a few times a day and look for tests, if you want one very soon. Sometimes someone cancels a test on short notice.

Also look around nearby testing centers. Maybe its worth it to take the train for 40 minutes, if you can get an attempt at another city?