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The 'Arnold Rimmer Song' is from the season 7 episode 'Blue'.
Dancing In Between by Amy Jo Johnson
The song is " Don't be fooled" by Hugh Dillon. off the Album "Works Well With Others"
The song is called "How Blue Can You Get."
The XX-Shelter
That song is Chances by the band Athlete.
The song was "One Step Closer". It was featured in the ER episode "On the Beach."
L'amour est Blue.. by Paul Mauriat (You are Welcome)
The song is "Bad Timing" by Blue Rodeo. It is off of their 1993 Album, "5 Days in July." This episode was actually the final episode of season 4, not the series finale.
The Barenaked Ladies play the song Pinch Me from their album Maroon during the episode but I don't know if this is the song you are asking about.
The song at the end of 'The Christmas Invasion' in Series 2, Episode 1 is "Song For Ten" from the Doctor Who: Original Television Soundtrack.
The song playing in the first episode of Red vs. Blue season 10 after the laser from the mother of Invention blows a hole in the ship is Terminate by Jeff Williams.