Hi Joe,
I’m not sure why your projects might be disappearing. Let me suggest a few things you could try. When you click Project, and Open, it opens another window. On the left, below Projects, Account, Options, etc. is box that you can enter a word to search for. If there are any letters or words in this box, Easel will only show you files that relate to that search query. Delete any letters or words in there, and it should display all projects.
The next item over to the right is Starred. If this is selected, the star goes full black, and will only show projects that you have starred. Click the Starred so that the star is hollow, as in just the outline of a star, not full black. Hopefully these tips will get you to be able to see all of your projects.
To answer your second question, yes, you can download your projects onto your local hard drive. Click Project, Download Project (ZIP), and it will create a zip file with all of the workpieces of your project. It will be saved in your default Downloads directory.
I have projects in Easel that date back to when I first started using it, so Easel does not just delete files after any period of time. Unless you have accidentally deleted them somehow, they should be there, right back to day one.