Hmm, the situation you describe is indeed most interesting…
The fact that your Home folder was moved may indicated that Mac OS X no longer treated it as such. Depending on the permissions of your "moved aside" Home and those of your current home, it is possible that the transfer of files did not happen as smoothly as one would have hoped — for example, the ownership of some system stored in our "Library" folder may be improperly set, prevent your computer from reading them.
While it is difficult to precisely know what may have happened without taking a closer look at your installation, there are a few steps you may want to take with regards to the problems you outline.
As far as the Finder sidebar goes, Mac OS X will automatically recognize you are dragging one of the predefined "Applications", "Music" or "Movies" folders there and assign it a system icon. Removing the item you dragged and adding it once again can sometimes force the Finder to recognize that icon. Should it not "stick", you may need to clear the caches of your installation, notably those in ~/Library/Caches.
As far as Help goes, looking at the Console log (available through the Console application, located in your Utilities folder) while the application is quitting or behaving in an unexpected fashion may provide you with interesting pointers. Usually, corrupt or missing help files are the cause of such unexpected quits.
Should the "weird" behavior persist, though, and given the serious shake that was inflicted to your installation, you may want to perform a clean-install of Mac OS X. While radical, this would effectively take care of any underlying issue we may have overlooked here.