The search bar has search options where you can type any information that can help you find an item.

For the "type" option you can search specifically by file type. Options include PDFs, Docs, Spreadsheets, Presentations, videos, etc.

For the "owner" option you can search for items that are owned by you or not. You can even search by a specific person if you know who shared it with you.

Don't delete anything from your "shared with me" folder. Consider it a landfill. You'll never be able to clean it up. If a student turns in an assignment through Google Classroom you own it until you return/grade it. If you own their document and you delete it, they also lose access to it.