You know that in order to get the two vowels before the five consonants, you must put the vowels in the first two spaces, so that you have a string of the form VVCCCCC. You can build such a string in two parts: first assign the vowels $A$ and $E$ to the VV slots, and then assign the consonants $B,C,D,F$, and $G$ to the CCCCC slots. You know that there are $2!$ ways to perform the first part of the task and $5!$ ways to perform the second part. These parts are independent: no matter whether you place the vowels in the order $AE$ or in the order $EA$, you can place the consonants in any of the $5!$ possible orders. So how should you combine the numbers $2!$ and $5!$ to get the total number of possible strings?
Your answer of $6!$ to the second question, however, is right, as is your reasoning: you can treat the pair $AE$ as a single entity, so that you really have only $6$ things to arrange, and there are $6!$ ways of arranging them.