You will be able to run both programs on the computer, but keep in mind that the OnRez Viewer and the Second Life Viewer are two different programs. Updates and patches are specific to each individual program.
No, when you run each program, the updates will be for the program you are currently running.
On a Mac, pressing command/Apple button while clicking the mouse does the same as using the right mouse button on a two button mouse.
Second Life performs much better on a direct connection, and may have bandwith/packet loss problems while on a wireless network. Use a direct (wired) connection wherever possible. Usually the problem with a wireless network is that connections can drop in and out.
Check both the OnRez and Second Life blogs to see whether it's a viewer issue, or in-world issue.
Your internet connection is likely very unstable, as when your packet loss is very high (the red bar in the top right corner) you tend to heat up the computer, which can cause an involuntary freeze or crash.
Restart your computer, and try closing any unnecessary programs to get the best experience and fastest response time in-world.