For a graph $H$, a graph $G$ is an $H$-graph if it is an intersection graph of connected subgraphs of some subdivision of $H$. $H$-graphs naturally generalize several important graph classes like interval or circular-arc graph. This class was introduced in the early 1990s by Bíró, Hujter, and Tuza. Recently, Chaplick et al. initiated the algorithmic study of $H$-graphs by showing that a number of fundamental optimization problems are solvable in polynomial time on $H$-graphs. We extend and complement these algorithmic findings in several directions. First we show that for every fixed $H$, the class of $H$-graphs is of logarithmically-bounded boolean-width (via mim-width). Pipelined with the plethora of known algorithms on graphs of bounded boolean-width, this describes a large class of problems solvable in polynomial time on $H$-graphs. We also observe that $H$-graphs are graphs with polynomially many minimal separators. Combined with the work of Fomin, Todinca and Villanger on algorithmic properties of such classes of graphs, this identify another wide class of problems solvable in polynomial time on $H$-graphs. The most fundamental optimization problems among the problems solvable in polynomial time on $H$-graphs are Maximum Clique, Maximum Independent Set, and Minimum Dominating Set. We provide a more refined complexity analysis of these problems from the perspective of Parameterized Complexity. We show that Maximum Independent Set and Minimum Dominating Set are W[1]-hard being parameterized by the size of $H$ plus the size of the solution. On the other hand, we prove that when $H$ is a tree, then Minimum Dominating Set is fixed-parameter tractable parameterized (FPT) by the size of $H$. For Maximum Clique we show that it admits a polynomial kernel parameterized by $H$ and the solution size.