When I try to install either chrome or opera, apper crashes. I have tried this several times and always encounter the same problem.

apper? Do you mean zypper? Or are you saying that the the application (Chrome or Opera) itself crashes?

...and how are you trying to install; are you using yast or zypper to install from standard (or other) repositories, or have you downloaded something and are trying to install that with the rpm utility?

I'm new to Suse, so I have no idea of what's going on, but if I double click the .rpm file, I get a query asking if I wish to install the browser in question, then it asks for my password. Then it tells me that I must download 10 other files. Then it resolves dependencies, and a few seconds later I get the message,"KDEinit could not launch usr/bin/apper."

So, you are doing the wrong thing, you are trying to use Linux like it was Windows and the end result is that the system nearly manages to stop you doing anything stupid and it nearly manages to work in spite of what you have done.

To do this correctly, go into 'install and remove applications' (name might be slightly different depending on which GUI you use), it will ask you for the root password, click on the application that you want to install and it will sort the problem out for you. There should be a very low risk of getting apps and libraries that don't match, so if you don't have a good of what you are doing, just stick to the approved way of installing things.

Don't try downloading individual rpm files, unless you know what you are doing, what the risks and the extent to which you are overriding the system's measures to protect you. It is safer and easier to manage in the long term.