I would like the designer to be able to open the same solution file as me. Is this possible with Expressions Web?
You can't open solution (*.sln) files, but you can open the folder containing them as a web site (using the "Open Web Site..." dialog), from which you can edit common web file types. You can also edit most text file types by right clicking and choosing Open With -> Expression Web (Open as Text).
The short answer is "no"; but this is a very common request, and I know a lot of people are hopeful that the next version (whenever that is) will have better support for this.
In particular, web application projects seem to be increasingly the default (especially for things like MVC, but obviously that changes the tooling a lot anyway).
You cannot develop Asp.NET or PHP applications with Expression Web. You can only use it to edit code files as text. This tool is a continuation of famous old Frontpage tool. It's much of a visual designer for html/css pages and templates