Thanks for asking.

Regarding question 1: 
Remote Buddy can't cause this by itself.

But I still think I know what happens. If you start EyeTV and Remote 
Buddy (or any other application other than EyeTV) at login time, 
chances are EyeTV first starts, goes fullscreen, hides title bar and 
dock. And then the second application (here Remote Buddy) finishes 
starting up and OS X makes it the active application. You should see 
that second application's name on the left in the titlebar.

Since the second active application doesn't ask OS X to hide titlebar 
and dock, OS X will now display them again. You should be able to get 
the same effect when you put EyeTV into fullscreen mode and then use 
Alt + Tab to switch to any other application currently running on your 
Mac.

Possible workarounds: 
1) Untested, may or may not show the respective effect: reduce Remote 
Buddy to a "UIElement" and adjust the "Prefs > General" settings so 
that Remote Buddy only shows as menu item, but not in the dock.

2) Remove the applications from your startup items, write an 
AppleScript that starts one application after another and add that 
script to your startup items. It could look like this:

-- snip -- 
tell application "Remote Buddy" to activate 
tell application "SomeOtherApp" to activate

delay 1 -- wait one second (shouldn't be needed, just here in case you 
need a delay somewhere)

tell application "EyeTV" to activate 
-- snap --

Regarding question 2: 
Remote Buddy will mute and unmute EyeTV when Front Row is entered/ 
left. EyeTV by itself will not do this.

Best regards, 
Felix Schwarz