I am not sure why you're having this problem. The normal way of getting apps onto the iPad is directly from the App Store app that is built into the iPad. I don't know why it's asking you to use iTunes. As you've found out, using the sync facility on iTunes is a real pain.
Don't worry if iTunes has deleted apps from your iPad. The App Store remembers which apps you've paid for and you can re-install them without paying.
A good 'fix' for any number of unexplained iPad 'problems' like this, is to completely switch the iPad on and off - rather than just, as most people normally do, putting it to 'sleep'.
To switch it completely off, as you probably know, you have to press *and hold* the power switch. Once the screen has gone completely blank - takes a few seconds - press *and hold* the power switch until the white Apple icon appears. It then takes about 10 seconds for the iPad to power up (during this time the screen orientation is disabled, so don't panic). Eventually the iPad plays a short 'jingle' and a couple of seconds later you're back in your 'Home' screen. Try using the App Store again and downloading the apps. See if the problem persists.
If it does, don't hesitate to come back to the Forum and I'm sure we'll have some other suggestions!!
Good Luck
Tim
Scotland