try 'sudo apt-get install libqt3', if it says it is going to to something which looks destructive, abort & post the messages apt is giving.
The 'remove a whole lot of applications' happens because dpkg (the utility which actually does the install) attempts to upgrade dependencies to match the new install which may mean that it will have to remove application X to upgrade it. As soon as it removes application X it also wants to remove everythinbg that depends on application X. It shouldn't work this way but sometimes it appears to want to.
Do 'sudo apt-get update' to update your package lists tothe latest before you attempt any installs. If apt-get update doesn't complete properly just repeat it until it does.