There are no designer files. For the specific error about <blog:searchonsearch> being an unknown server tag, one thing to make sure is that SearchOnSearch.cs exists in the App_Code\Controls folder.
A second thing is that all these files work in a Web Site project. If you try to convert BE into a Web Application project (WAP), it's possible this type of problem could occur. You mentioned "designer" files. Web Site projects
don't have designer files ... WAP projects do have them.
It's also possible that VS may initially report that <blog:searchonsearch> is an "unknown server tag", but you might want to try building or running it (F5), and then VS will hopefully start detecting SearchOnSearch.
Files in the App_Code folder, like SearchOnSearch, are compiled at runtime. They are then available to be used. I may have seen a circumstance where until the project gets built once, files in the App_Code directory are not recognized.
When you say that you're starting a "blank project" ... what type of project is that? A Web Site project or Web Application project?