Recovery of a deleted APP

All apps on your iPhone should be in your iTunes library on your computer. If you have Sync Apps selected under the Apps tab for your iPhone sync preferences with iTunes, after downloading an app with your iPhone the app will be copied to your iTunes library on your computer the first time you sync your iPhone with iTunes after doing to. Unless the app was deleted from your iTunes library on your computer, this allows for re-installing the app on your iPhone by making sure the app is selected under the Apps tab followed by syncing your iPhone with iTunes. This also allows for removing an app from your iPhone by deselecting the app under the Apps tab followed by a sync. To reinstall the app later, reselect the app under the Apps tab followed by a sync.

All apps can be downloaded an unlimited number of times. You only pay the first time.

Download it again on the iPhone. It will ask you first if you want to buy it, but later it will tell you that it's a free download because you already paid for it.

Then, sort out your sync issues. Buy OS X Leopard (a bit hard to find) for your current iMac (and make sure it has at least 1 GB of memory), or a newer computer. You're asking for trouble by not syncing regularly. While you can download apps as many times as you want, music and videos can only be downloaded once.