Screen items do not stay in place!

I use Mission Control and desktop spaces with no problems. My apps open consistently where they are usually assigned across four or five different desktops, depending on how many apps I have open.

If you currently have apps assigned to "None", then they will open in whatever desktop you happen to launch them on.

Applications will open consistently in their assigned desktop/screen and windows will remain in place on their assigned desktops IF you assign them to particular desktops.

Start by creating however many desktops you need with Mission Control. To help differentiate the desktops from one another, change the Wallpaper on each.

You can now launch your apps, one at a time, and assign them to their own desktop. The application windows will open and remain on the desktop the app is assigned to. Use the "Assign To" option of the apps Dock icon to assign your apps.

If you Assign To: This Desktop then the app will always open on that desktop.

If you Assign To: All Desktops then the app will open on whatever desktop you happen to be in when you launch it, and its windows will also open on that desktop. And when you move to a different desktop and recall the app it and its windows may follow you there.

Also very important, to ensure switching to an app opens it on the desktop where it has open windows, go to System Settings > Desktop & Dock, click Mission Control, and enable the "When switching to an application, switch to a space with open windows for the application" setting.

Work in multiple spaces on Mac - Apple Support

You're welcome.

Mission Control and spaces can be a little confusing to get set up initially, but once you get it dialed in it can be indispensable.

I don't know why Mail should be quitting unprompted.

But I do have Mail, Pages, Numbers, Safari, Music, Logic, News, Weather and several other apps open all the time on my Mac. They all behave well within the parameters of Mission Control and don't give me any problems. When I shutdown and restart, the apps and their windows stay put. I've been using these features for many years now, though it did take me a while to warm up to it.

I think you might try assigning Word to "All Desktops" so that those files don't all stack up in the same desktop. The same goes for Preview (or whatever PFD app you use) if you need to have .pdf files open on several desktops. I *think* that is the appropriate assignment - though it may actually be "None". Try either/both and see what you come up with.

I should also say that if you happen to be running any anti-virus, clean-up, optimizing or VPN software it may be that there is some conflict with the OS from any of those. Apps of the type very often create behavioral problems on the Mac. You'd be wise to uninstall any of those and keep that junk off the computer. They are unneeded.

You’re welcome.

Good luck. 🍀

Don’t hesitate to post again if you need anything additional.