Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?
I think the answers already given by Thomas Kalinowski are more comprehensive, but Douglas B. West maintains another collection of open problems in graph theory at http://www.math.illinois.edu/~dwest/openp/ and there are also several graph theory problems listed at https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Graph_theory
See Graffiti by Siemion Fajtlowicz. But the great majority of these problems (mainly on graph theory) were not by Fajtlowicz directly but by Graffiti itself (only some were jointly obtained by Graffiti and Fajtlowicz), while Graffiti is a computer program created by Siemion. (Paul Erdos liked the Graffiti conjectures, so you may too). :-)
Here is a nice problem about graphs: it is true that every Cayley graph of every finitely generated cancellative semigroup must have either $1$, or $2$, or $\infty$-many ends (number of connected components looked from the infinity perspective, roughly speaking). But yet, no analog of Stallings Theorem for finitely generated cancellative semigroups is known. It would be fun to know.