Pedersen commitments, what happens if I choose $H$ such that $H = a\times G$? For Pedersen commitments of the form $C = x\times G + r\times H$, what is the worst thing I can do if I already know $H$ such that $H = a\times G$ ? 

For standard curves, there are specifications for what value of $G$ should be used. Is there any such specification for values of $H$ ? What is the worst an adversary can do when they choose $H = a\times G$ and the rest of the world assumes that no-one knows a relation between $H$ and $G$ ?
If the committer knows $a$, then it is no longer binding. 

For any $m$ you commit, the commitment will be $(m+a)G$, so it is easy to find $m'$ and $a'$ such that $m'+a'=m+a$ to open the commitment.

Edit to add:
If the receiver knows $a$, then it is still information theoratically hiding because the sender chooses $r$ which is uniformly random.