The group $G$ is proved infinite in the paper

M. Edjvet and A. Juhàsz, The groups $G^{m,n,p}$,
J. Algebra 319 (2008), 248-266.

The proof is geoemtric using pictures (which are similar to van Kampen diagrams), and doesn't provide any information about the quotients of $G$.

As far as I know, ${\rm PSL}(2,43)$ is the only known finite quotient. I checked this for all simple groups up to order $10^9$, and also that it is the only ${\rm PSL}(2,q)$ that is a quotient. The kernel of the homomorphism onto ${\rm PSL}(2,43)$ is perfect, so there are no further quotients to be easily found there.

For your other question, I got one further than you did, and proved using a big coset enumeration over the subgroup $\langle abc \rangle$ that the quotient is trivial for $i=17$. I am trying $i=18$, but I am not sure where this is leading, because you will inevitably get stuck at some point.