Dear Mauro, Helmut,
I’ll be focusing on logical graphs, especially the duality between entitative and existential interpretations, for quite a while longer, so this doesn’t address your questions about modal logic, but you might find it useful to compare the representations of logical operators by means of truth tables with those using logical graphs.
You could start with the top eight entries in the section headed “Logical Operators” on the following page.
There’s also a page bringing all eight of those Truth Tables together in one place.
I had been meaning to include the corresponding Logical Graphs and Venn Diagrams — I’ll spend some of my pandemic time working on that — It looks like it would be worth the candle reviewing their properties as representations of basic operations and going over their relative utilities for various logical purposes.
The following two pages also contain useful synopses of the boolean basics.