I don't know the root of the problem is, but I can list some things that it isn't.
First, it isn't SOP. I've never seen that problem, neither on my machine, nor mentioned in any forum I read, except in this thread.
The fist time you drag something to the desktop, it goes to one place, but the second and later times it goes somewhere else. Ergo, it's not anything to do with your View Options.
Each folder has its own .DS_Store file. Each user has their own Desktop folder, with its own .DS_Store file. A new user would have a new folder and a new file, both uncorrupted and with correct permissions. If the problem occurs as described with a new user, it's not a .DS_Store issue.
On MacOS, there was a separate invisible Desktop folder on each disk volume, and Finder merged their contents on the fly to put all their icons on the desktop. That caused numerous problems, some of which are similar to the ones described. (You could not easily tell which volume an item on the desktop was actually on. If you dragged it to a folder, even a folder on the desktop, you were implicitly dragging it to whatever volume that folder was on. If that meant it was going to a different volume, you got a copy. Otherwise, you got a move.) But OS X doesn't handle the desktop that way. Instead, each user gets a prominently visible Desktop folder right at the top of their home folder, and whatever's in that folder is what they see on the desktop. There's no longer any merging of folders from different volumes. Besides, if I understand your complaint about being asked to buy an external disk for backup, you don't have any other disk volumes anyway. So that's not it.
Notice that I'm scrupulously distinguishing between "Desktop", with a capital "D", and "desktop" with a lower-case "d". "Desktop" is a folder, not much different from any other folder. "desktop" is the area on your screen behind all your windows. The connection between them is that they contain the same items. Whatever you put in one appears in both.
"Desktop", like any other folder, can be viewed in Finder through a window. In fact, you can open any number of windows into the same folder, and they'll all show the same items. (Use ⌘^O to "Open in New Window". Or just use ⌘N repeatedly to get several windows, then navigate each of them to the desired folder.) MacOS couldn't do that, because it failed to distinguish between a folder and a window into that folder.
Each window can be put into one of four "views" (List View, Icon View, Column View, or Cover Flow View). Notice that view is an attribute of a window, not a folder. Each of these views has its own independent "View Options". The desktop itself can be thought of as yet another window, differing from other windows in that it always looks into the Desktop folder, and is always in a fifth view (Desktop View), which naturally has its View Options as well. (The five views don't have completely independent options. Desktop and Column view do, but there's a lot of crosstalk between List and Coverflow, and both of those share Arrange/Sort by... with Icon view.)
When I look at the View Options for Desktop View, there is no "Arrange by ..." option. (This is one of the differences between Desktop View and the very similar Icon View.) Try as I might, I cannot find any option that makes new icons appear in the upper LEFT of the desktop. If "Sort by..." is set to anything but "none", icons arrange themselves starting from the upper RIGHT.
If I open a window on the Desktop folder, and put that window in Icon View, and set Icon View's View Options to "none"/"none", then new items appear in the first available position in that window starting from the upper LEFT. Is that maybe what you're seeing? Do you have a window open into the Desktop while you're doing your tests? The Desktop and desktop are using different views, and while they're displaying the same items they will generally display them differently. But that doesn't get the icons stacked up on top of each other, so I guess that's not it either.
BTW: You really do need that external drive for Time Machine. They're cheap nowadays. You can get gobs of disk space for under $100. Without backup, you will someday lose all your data, including the irreplaceable stuff.
