How to lock icon position on desktop

Click on the Desktop then select Show View Options from the Finder's View menu. Select the appropriate Arrange By option from the drop down menu. Snap to Grid will let you arrange the icons as you like but maintain a pre-set grid layout on the Desktop. None will let you put them anywhere without any pre-set grid layout.

I agree that the icon arrangement ought to persist through a reboot; AFAIK the Finder is supposed to save that information before shutting down.

Just a shot in the dark: is there a .DS_Store file on the Desktop? (It would be invisible, so you’d need to use Terminal or a third-party utility to look for it.) If you find one, delete it. Another possibility is a damaged preference file. Have you verified the disk & its access permissions with Disk Utility?

They will stay where you put them.

They don't stay in the same place once you re-boot your computer though. Is there a way to fix that?

I don't know what to suggest. Works on my computers.