Open problems related to Graph isomorphism
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 [2], has been asked here), maybe solved: http://arxiv.org/abs/1404.0818
cutwidth / bandwidth [1]
treewidth-k vertex deletion set size (feedback vertex set number in [7])
tree / path distance width (see [1]), connected tree distance width (see [3], 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 [4])
maximum degree [5]
genus [6] / crossing number [8]


Note that there is active ongoing research for some them.

[1] : K. Yamazaki, H. L. Bodlaender, B. de Fluiter and D. M. Thilikos. Isomorphism
for graphs of bounded distance width. Algorithmica 24.2 (1999)

[2] : H. L. Bodlaender. Polynomial algorithms for graph isomorphism and chro-
matic index on partialk-trees. Journal of Algorithms 11.4 (1990)

[3] : Y. Otachi. Isomorphism for Graphs of Bounded Connected-Path-Distance-
Width. Algorithms and Computation. Springer, 2012

[4] : http://www.fi.muni.cz/~hlineny/res-en.html#recent

[5] : L. Babai and E. M. Luks. Canonical labeling of graphs. STOC '83.

[6] : 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 

[7] : S. Kratsch and P. Schweitzer. Isomorphism for graphs of bounded feedback vertex set number. SWAT 2010 

[8] : http://math.mit.edu/news/summer/SPURprojects/2012Velednitsky.pdf