My iPad 2 won't transfer to PC!

Question

Ok, pulling my hair out now after searching all these forums.  Trying to update my iPad 2 from 5.1.1 to 6.1.3.  iPad is nearly fully, and in fact states in iTunes that it is over capacity by 215.7 MB (on a 16 GB Pad). 
I am having the following problems/pains(!):
1.  When I look at the list of apps in iTunes, it states there are 105 apps, but when I count all the apps in the list next to the iPad screenshots, there are only 58.  i.e., the screenshots accurately portray what's on the iPad; the list to the left does not.
2.  When I try to update, iTunes tells me there are purchases on the iPad that I should transfer.  So, I dutifully attempt to do this, and it seems to transfer 67 files (which is obviously not the 105 that iTunes is telling me are installed!); chokes a bit on the first two files, I think they're the 1.x GB files, and then quickly goes through the rest of the 67 (not 105).
3.  Enter 1 Infinite Loop...  Assuming the transfer is a success, because I've not been told otherwise, I try to update again... And again am informed there are purchases that I need to transfer...  Loop... Pull Hair out... Post here... Have everyone tell me to look at articles about backing up and restoring.  Those don't work. 
Also, what good is it to restore settings from a backup, if the backup has bad settings?  Is this really a solution?
The main thing I'm interested in recovering is the Apps, which obviously aren't transferring, nor do they seem willing to...  I suppose I can download them all again one by one from "the cloud", but then aren't all the app specific settings lost?  Or is that what backup is supposed to take care of?  Even though according to iTunes I only have 58 of 105 applications?!?  Any why is the iTunes "List" (next to the screenshots) listing 58 apps, while the "transfer purchases from iPad" action seems to be finding 67?  I'm so confused as to how all this is interacting, why memory issues on the iPad are preventing purchases from being transferred FROM the iPad properly, why there are no error messages after a "transfer purchases" attempt.
As one other piece of interesting (useless?) information, iTunes shows that I have 3.2 GB of "other" data.  Not sure about the importance of this, but I think before I started any kind "transfer of purchases", it was only at about 1.5 GB or so...  That's just an approximation, but the point is that the iPad wasn't maxed out of memory until I began all of this.
Any help appreciated, but I'm inclined to believe will not be forthcoming.  Thanks anyway to anyone that was brave enough (bored?) to read through all this dribble! 
I'll nominate anyone for 1 billion points if they can solve this.  I've not found anything that even comes close to addressing this issue.

Answer 1

What number shows in Settings > General > About > Applications on your iPad, does that tie up with the 105 (is that the number shown at the top right of the Apps tab above the screen mock-ups) or the 58 (or is it a third number) ? The figure in Settings and at the top right of the screen should be the number of downloaded apps, so it excludes the built-in apps. The list of apps to the left of the screen mock-ups are the apps that are in your computer's iTunes library which are compatible with your iPad - so some might have an 'install' button next to them if they aren't currently on your iPad, those with a 'remove' button should already be on the iPad (so the 'remove' button would delete the app, and its content, from the iPad).
When you try the File > Devices > Transfer Purchases that isn't just looking at apps, it should also transfer any iTunes music, film, TV show, audiobook and ibook downloads over from the iPad (it should also copy any PDFs or epubs that you have in the iBooks app even though they aren't iTunes purchases) that aren't already on your computer - so the 67 is the count of items that aren't already on your computer that it should copy over, it will exclude any apps (and other iTunes dowoloads) that are already there, and it could also include other things.
The 'transfer purchases' also only copies the actual apps, it doesn't copy any documents / settings / game progress etc that you have in those apps i.e. it's just the app as downloaded from the store - their content and settings should be included in the backup instead. Before starting an update I normally advise people to take their own copy of any important documents that they have on the iPad, just in case the backup gets corrupted and can't be restored to the iPad e.g. via the file sharing section at the bottom of the Apps tab, emailing, transfer via wifi (depending upon what the app supports).
You have enough free space on your computer, are there any items that you could safely delete from the iPad to see if freeing up space might get the transfer to work, and/or have you tried rebooting your computer ?