I'm going to go out on a limb here and guess at what you mean.

When you say that you can't drag icons to a new spot, I am going to assume that when you try to do that, and the icon drags away, but when you release the mouse (left) button, the icon snaps back to the left area of the desktop.
If I described pretty much like what is happening for you, it is because your desktop property is set to "Auto Arrange"
To change that behavior just right-click anywhere on the clear area of your desktop, select "Arrange Icons" and you should see a checkmark next to "Auto Arrange." select that, which shuts off (un-checks) the Auto Arranging function, and gets the action you probably want.

As for the action you are referring to as "single-click" and "double-click" mode, here again I am going to assume something. I assume you mean that when you single click on a desktop icon, that the icon opens, and what you prefer to happen is that a single-click simply selects the icon, and you also wan, instead, two rapid clicks to actually open the icon.
If I have correctly assumed what it is you want, then this behavior is generally caused (but not only) by setting your desktop to use the "Active Desktop," that is where you can have your desktop viewed like a webpage. And if you can think of it, any link on webpage just requires one-click to activate it. Thus, having your desktop perform as a webpage (or something like a webpage) says you want a single click to open icons.
However, turning off the desktop Active Desktop function, will not necessarily turn off that single-click function.
To turn off that single-click function (whether or not you are using the Active Desktop, just open your Windows Explorer, select Start>Programs>Windows Explorer (or you can press and hold the Windows "flag" key and press the letter "E" then release both keys).
In the Windows Explorer, select Tools>Folder Options, and on the General tab, select the Classic button, and the [Settings] button. There you will see the setting you want "Double-click to open an item"

That's all! Have fun.