Database recovery procedure I recovered a database with last full backup, all differentials after the full, and then all log backups after the last differential.

Can I trust the database is right or do I have to do it again the right way: with full, then last differential, then all transaction backups after that?
No, you dont have to do it again.

The differential backups are just that so the only effect your mistake made was to make the restore time longer than neccessary. If your last transaction log backup was restored successfully then you are ok.