1) You can configure the menu to your liking at Preferences > Menu. 
The built-in ones can't be modified, but you can easily create a new 
one (which can also be a copy of a built-in one) by pressing the "+" 
button at the down-left in Preferences > Menu. Please find step-by- 
step instructions in the Help's "How-To" section.

The trial version has no feature limitations at all. It's really just 
time limited to 30 days.

2) You can add Custom Actions (=> again, please see the How-To section 
for instructions) that contain an AppleScript to quit a specific 
application. For EyeTV, the required AppleScript code would be

tell application "EyeTV" to quit

3) Remote Buddy automatically selects the correct Behaviour for the 
currently active application. If EyeTV is the active application, that 
will be EyeTV. If Front Row is the active application, that'll be 
Front Row. I'm afraid that pyetv works in a way that breaks this 
concept. You should be able to use Remote Buddy's virtual remote, 
though, to simply pass through any remote events as Apple Remote 
button events, so that - as lng as the virtual remote is active - 
you'll get the exact same behaviour from your remote as without Remote 
Buddy - which may be desirable in the case of pyetv.

4) Regarding Remote Buddy: does it really crash? (requester coming up) 
Or does it get unresponsive? (beachball). The latter - if happening 
after an EyeTV crash - is most likely an AppleScript targeting EyeTV 
being executed by Remote Buddy and waiting for a response from EyeTV - 
which - if EyeTV crashed - will be forever.

Please contact elgato support regarding crashes of eyeTV.

Apart from that, I'm afraid that your plan to store live recorded 
video on the AirPort Extreme Base Station will not work, because its 
performance is (with high likelyness) too low to keep up with the 
amount of data that even standard definition television video is 
causing every second and that needs to be written away. An internal 
harddrive can easily cope with that amount of data, but a network 
drive may easily be struggling.

Best regards, 
Felix Schwarz