a) How can I connect two separate verticies and create a new edge?
So there are two tools, Append Polygon, and Append Edge. Append Edge is the one you want.
Just select Append Edge, then click one vertex after the other. Right click to Confirm the command.
b) Is there some vertex sliding possible?
You can move a vertex along an axis, I have not found a way to ‘slide’ a vertex along an edge, but you can slide edges along a face.
c) How can I just create freely new verticies in an empty space?
I use the Append Edges tools here too, to create free paths. Making a single vertex does nothing, but two or more creates an edge or path of edges.
d) Is there some functionality to distribute verticies evenly on a edge?
You can remove all of them and use the Cut tool. Set the number of sections in the ‘opt’ dialog first then select the edge you want to distribute along and click Cut. If you select Edges in a line (loop) each will be cut. If you select them parallel, in a ring, you will make a new edge at the cut.
Hope this helps!