Does a full transaction log cause a rollback? Will a transaction that is trying to write something into the transaction log be rolled back if the transaction log is full?
Yes but it will be the rollback which actually causes the log to become physically full.

As a transaction generates log records, it will also reserve space in the log for any records required required to rollback. With 20MB of log space remaining and a hypothetical transaction that generates 10MB of log records, you would receive the log full error with 10MB still free and the rollback would then consume that 10MB.