How to join meshes with some peculiarities

I’m trying to develop a product that will eventually be generated from several meshes, but in the end it needs to behave as a single object.

I’ve already created the first mesh, yet when I try to merge a second mesh exactly where I want it, I can’t get it to work.

Second question: If I start with an existing mesh but later want to keep only some of its vertices, how can I build a new mesh from those vertices? (Using Cull Faces hasn’t given me the result I need.)

Thanks in advance for the help!

It helps to post at least some screenshots…

One way to combine meshes is by shrinkwrapping everything