Try resetting the preferences. To restore preferences quickly using a keyboard shortcut:
Press and hold Alt+Control+Shift (Windows) or Option+Command+Shift (Mac OS) as you start Photoshop. You are prompted to delete the current settings. The new preferences files are created the next time you start Photoshop. Note: Using the keyboard shortcut, preference files for custom shortcuts, workspaces, and color settings will also be reset to default.
This does not happen on my Windows PC. When photoshop is closed down all Photoshop windows and palettes are closed down. When you open Photoshop the Layer Palette should be on the display it was on if the layer palette was currently being displayed when you closed down Photoshop. If you move it it show stay put where you moved it. Resetting your workspace will mover to where you set it up in your work space. Open or closed.
Do you mean close the documents or quit Photoshop? When you close documents, any layers that are floating will appear in front of the Welcome Screen. Try Windows > Workspace > Reset [Current Workspace] and to dock all the panels and see if it still happens.
If you are quitting Photoshop, all panels should close. Check to see that the application is really closed, or close it this way and see if it still happens:
Mac: Photoshop menu > Quit
Win: File menu > Exit (I think)
We might need to see a screen shot to better assess.