For the first question: Graph Isomorphism has been considered for at least the following parameters for which fixed-parameter tractability is still open.
- pathwidth / treewidth (see , has been asked here), maybe solved: http://arxiv.org/abs/1404.0818
- cutwidth / bandwidth 
- treewidth-k vertex deletion set size (feedback vertex set number in )
- tree / path distance width (see ),
connected tree distance width (see , however you can get quite close to the last one, see section 6.4. of my diploma thesis ) : solved by Y. Otachi and P. Schweitzer : http://arxiv.org/abs/1403.7238
- clique-width / shrub-depth (or SC-depth) (see )
- maximum degree 
- genus  / crossing number 
Note that there is active ongoing research for some them.
 : K. Yamazaki, H. L. Bodlaender, B. de Fluiter and D. M. Thilikos. Isomorphism
for graphs of bounded distance width. Algorithmica 24.2 (1999)
 : H. L. Bodlaender. Polynomial algorithms for graph isomorphism and chro-
matic index on partialk-trees. Journal of Algorithms 11.4 (1990)
 : Y. Otachi. Isomorphism for Graphs of Bounded Connected-Path-Distance-
Width. Algorithms and Computation. Springer, 2012
 : http://www.fi.muni.cz/~hlineny/res-en.html#recent
 : L. Babai and E. M. Luks. Canonical labeling of graphs. STOC '83.
 : I. S. Filotti and J. N. Mayer. A polynomial-time algorithm for determining the
isomorphism of graphs of fixed genus. STOC '80 /
G. Miller. Isomorphism testing for graphs of bounded genus. STOC '80
 : S. Kratsch and P. Schweitzer. Isomorphism for graphs of bounded feedback vertex set number. SWAT 2010
 : http://math.mit.edu/news/summer/SPURprojects/2012Velednitsky.pdf