Why don't my iPad apps show in iTunes?

The list will only show the apps that you have in the Apps part of your Mac's iTunes library. You should be able to copy any apps that aren't there from your iPad via File > Devices > Transfer Purchases, or download them directly on your Mac by clicking on your name/icon towards the top right of the iTunes screen and then clicking Purchased on the popup :

The two sections, 'My iPad Apps' and 'My iPhone Apps' are just to distinguish which of the apps are designed for iPads and/or iPhones (and iPod Touches). The same app can show in both i.e. if they are 'Universal' apps (they are coded for both iPads and iPhones ; they have processing in them to make the best use of the device's screen), some will only show in 'My iPad Apps' (they are designed specifically for iPads and can't be installed on iPhones), and some in 'My iPhone Apps' (they are designed for the smaller iPhone-sized screen ; the majority of them will work on iPads, except those that require full phone capability ; when used on an iPad they should have a '1x' or '2x' button on the screen to change how much of the iPad's screen they use).

You can turn auto sync off and when you connect the iPad to the Mac, iTunes will not launch nor will the device sync on its own. When you do initiate the sync or backup process, you should get a dialog box that will pop up and alert you that you have purchases on the iPad that need to be transferred into your iTunes library.

Launch iTunes (don't connect the iPad yet) and go to iTunes>Preferences>Devices>Prevent iPods, iPhones and iPads from automatically syncing. Click OK. Now you can connect the iPad and start the sync when you are ready to do so.

The apps and your backup are separate, and both are needed to be able to restore to a backup. The backup itself doesn't contain any of your downloads (apps, music, films etc), it contains things such as your photos, documents, settings, Safari cookies and history etc - the restore will look to pick up the apps (and your music, films etc) from your iTunes library, so for it to be able to work fully you will need all of the apps (those that were on the iPad when the backup was taken) in the Apps part of your computer's itunes library.

If, with the iPad connected, you do File > Devices > Transfer Purchases in iTunes that should copy over to your computer any apps (and any other store download) that aren't already in the library.

Both Transfer Purchases and backing up should be done regularly, especially if you are important documents/files that you work on on the iPad, if the iPad is lost, stolen, or breaks you will lose anything that you've worked on since your last backup. Apps (and music etc) can generally be redownloaded from the store (via the Purchased tabs in the store apps), but the content provider/rights-holder could remove an item from the store at any time, in which case you wouldn't be able to redownload them - so having your own backup on your computer (and also preferably on a backup drive) is useful.