Welcome to the discussions,
did you try to download the app using iTunes on your computer? You can download apps for free again:
http://support.apple.com/kb/HT2519
Data created by apps is part of the backup, apps are not included in an iPhone backup. See what's inside here:
http://support.apple.com/kb/HT1766
Syncing to another computer will erase the content on your device and replace it with the content of the new iTunes library.
To be able to save your data and settings, back up manually before:
If you haven't already, disable autosync in iTunes/settings/devices, connect the phone, right click on it in the device list and choose backup. iTunes will perform a backup without syncing.
Try to transfer your purchases again by choosing "transfer purchases".
Make sure to have at least one contact and event listed on your computer, to be able to merge contacts and calendars during the sync process. After your phone got wiped and set up again, restore from the manual backup you made and all settings and data should be present again.