Vertex snapping with axis constraint (like in 3ds Max) In 3ds Max there is an option for snapping with axis constraint so I can snap like in the image on the right side. Can I do this in Blender? In this case custom orientation isn't working, only way I can solve this case is cutting with Knife Tool and angle constraint enabled.
Also is there an option for snapping to the edge, constrained to one axis (with snapping enabled it's always sliding along edge so I can't get precise results)?
Snapping system in Blender leaves a lot to be desired in terms of precision modelling at the moment, unfortunately.

One possible solution is to use the Tiny CAD addon (bundled with Blender by default) that has some CAD like intersection tools to create vertex at desired edge intersections.

Here is a workaround for your specific illustrated case without resorting to addons.

Move the vertex vertically until it lines up with the desired snap target, regardless of edge direction, then move it to the side so it forms a convex shape in relation to the two end vertex from said edge.

Select both ends and press J to create a straight edge running between them.

You can now either delete the extraneous vertex, or use the merge tool at Alt + M > At Last to merge the undesired vertex into place.