This might not be your problem but it's the only thing within Maya that I can think of that would cause such an effect and that is the selection masks on the Status line ( thats the second row down of icons on the GUI), if you have activated one of these then you may not be able to select and do certain tasks. F8 toggles between object mode and component mode so also make sure you are in the right mode when trying to perform a task.

It is also possible to template an object by right clicking on it and under actions there is a command to template and untemplate but that would also make the object unselectable and I gather you can select components but simply can't manipulate them.

Failling that it's a bit of a mystery and might be something like a graphics driver issue or even a faulty program install as this issue seems almost unique to you.