I wasn't getting any new work from E@H on my GPU computer

When running more than one project one should not expect there to always be work for all projects in the cache at all times. It all depends on how you have your "Resource share" setup between projects and if your projects have been able to provide work when Boinc on your computer requested it.

If you open Boinc Manager on your computer and look in the "Projects" tab of the "Advanced view" you should have a column named "Resource share". Having big differences in resource share makes it harder for Boinc to try and live up to your wishes and if projects can't deliver work when requested that also makes things worse.

Best advice is to set a fairly low work cache when running multiple projects and then just let it run, it should even out somewhere in the region of your set resource share, but it will take time and sometimes that can be in the order of weeks or months.

Einstein is fairly stable and most of the time you get all the work you request compared to Seti that has been more unstable and less prone to fulfill work request, I would guess that your computer had done more Einstein work than your resource share was set to and so was trying to catch up with Seti work. This might also be true for other projects you are running.
When the newly downloaded Einstein work is done I suspect that Boinc on your computer will try to catch up on other projects for a while, so you should expect there to not be any Einstein work in your cache for a while.
If you change you resource share things will be different and if you start to suspend projects things will probably be worse than just letting it run and sort itself out.

So you run 6 projects. Then I would advice you to have a very small work cache setting, something like 0.5 days total. Boinc should always ask for work from the project that has done the least amount of work in recent time first. The more often Boinc asks for work the greater the chance of that request going through. That will help Boinc to honor your set resource share.

With so many projects active the chance for all of them being unable to provide work and you running out of work to do seems quite remote, unless your Internet connection is unreliable.

That's good, keep it at 0.1, increasing the cache size will only make it harder for Boinc to balance the resource share. 

Boinc probably thinks you've done to much Einstein work and are trying to catch up on other projects. I suspect that one or more of the other projects have had problems delivering work a while ago so Boinc ran work from Einstein instead, that debt is now being repaid. If you leave Boinc to do it's thing for a few days/weeks it should go back to normal and alternate between your different projects.

This shouldn't have any effect on what projects Boinc thinks should run due to resource share.
But mixing projects with both GPU and CPU apps with CPU only projects will probably lead to the CPU only projects dominating the CPU work on the host while the GPU/CPU project will only use the GPU.