The pencil is the standard icon for all editing function in DNN and you'll see it all over the place. In the case of the links, the image is only visible to a person signed in with edit rights on the module. It allows you to edit that link again. Your visitors will not see it. Note that there is a bullet image option in the links module settings page. You can specifiy an image that will be visible to your visitors in there.
You can however also replace the edit.gif file in the \images folder with one of your own, but note that you must replace it again if you upgrade your portal, and note that you're the only person who sees it anyway.
The Events module is one module that should be upgraded to the latest version 4.x as hte previous versions are quite buggy. However, wait until that version has come out of the release tracker where it is being tested. There will be an announcement on this site.
When time comes to upgrade, most modules are simply uploaded via the Host, Modules page, but always look for release notes. Modules are packaged as zip files called Private Assemblies or PA's. Have a look iside the module zip to ascertain that the file is the actual PA, and not holding one or more PA's inside it. Download and look inside a few modules to become familiar with their contents.
In all cases, you should do one thing before doing anything else: You must sort out a backup and replace procedure for your file system and database. If you ignore this advice then be prepared to deal with some pain further down the road.
Regards,
Rob