My family has 3 iPhone 5's and we share one iTunes account. We sync on 2 different computers. I do not want all the same apps on my phone. Can I delete them from my phone without it deleting them without screwing up the other 2 phones?
Yes - deleting an app from any device will not remove it from other devices.
Only if you:
- Delete the app from iTunes
- Sync more than one device to iTunes
- Bypass the warning that an app is present on the device but not iTunes and skip the chance to import that app back into iTunes
will the deletion of an app in one place cascade over to other devices. You might also want to review which devices automatically download apps so that you don't get apps that are purchased and intended for another device.
Also, be sure to disable automatic App downloads on your iPhone by going to Settings > iTunes & App Store. This way if someone from your family buys an app on their iPhone, it will not automatically download to your iPhone.