Problems with syncing apps

Question

Recently my computer crashed so I restored it to its original state. Now, when i try to sync my iPhone the apps don't sync. On the App page in iTunes there are no apps listed on the left side of the screen and the screenshot of my iPhone is somewhat grayed out so I can't move apps around and i can't even select a different page to view.
The check box by Sync Apps above where the list of apps should be is unchecked. If i check the box it says that the apps from the computer will overwrite the apps on my phone and since there are no apps on the computer it appears like it will delete everything except the original iPhone apps. All the homescreens (on the screenshot on the right side) go blank - as though everything will be deleted - except for the original apps.
Where are the apps supposed to be backed up on the computer? I thought I used to have a fairly easy way to access the folder where the apps are synced to but i can't remember how to get to them. In my iTunes library there is a folder called Mobile Applications where all my apps are stored (since my last sync before my computer crashed) but they don't show up in my apps list in iTunes. Is there some way to make them show up or are they stored in the correct folder?
Or, if I set the check box to Sync Apps will that copy my apps from my iPhone to my computer that aren't saved on my computer AS WELL as overwriting ones that are on my computer (which is none as far as iTunes is concerned) onto my iPhone? There are a few paid apps that i have deleted from my iPhone but are backed up on my computer. How can I get them back since it doesn't seem to be syncing properly and iTunes isn't recognizing the folder that they are in?
I just don't want to end up having most of my apps deleted and can't get them back so before i go any further I would like some input on this.

Answer 1

All 3rd party apps on your iPhone should be in your iTunes library on your computer, which is the same for all iTunes content on your iPhone. Your iTunes library on your computer - the iTunes named folder in your Music folder should be included with your computer's backup along with all other important data on your computer's hard drive. The iPhone is not designed or intended to be used as a backup storage device or as an external drive for transferring data from one computer to another.
Did you lose all important data on your computer's hard drive due to this crash, or your iTunes library only?
The sync process for iTunes content is a one way transfer process only - from your iTunes library on your computer to your iPhone, and the same for photos transferred from your computer.
In regards to iTunes content, an iPhone can be synced with an iTunes library on a single computer only, and photos can be transferred from a single computer only. When transferring iTunes content and photos from another computer, all iTunes content and photos transferred from a different computer will be erased from the iPhone first.
After restoring your computer to its original state, your computer is detected as a new or different computer. This would not be a problem if you maintained a backup for all important data on your computer which should include your iTunes library stored on an external drive or some other external media used for backup purposes only.
iTunes includes an option to transfer iTunes content that was purchased/downloaded from the iTunes store from an iPod or iPhone, but this is for iTunes content that was purchased/downloaded from the iTunes store only.
First you need to re-authorize your computer with your iTunes account with iTunes. With your iPhone connected to iTunes and without syncing, at the iTunes menu bar go to File and select Transfer Purchases From - the name of your iPhone.