There are a large number of songs with the word "dog" in the title. Listed bellow, are a few. Note that this list is not inclusive.
The title of the song is Footlights.
Unfortunately no. You can however make a parody of that song title. And you might be able to if the song title is a generic word.
With a Song in My Heart (1952)
Golmaal Is Back Again Title Song 2011 This is all I could find...
£6.99 with sweet and sour sauce
Angie is the title of a song recorded by The Rolling Stones. Alfie is the title of a song recorded by Dionne Warwick.
For what its worth but Buffalo Springfield
glass onion
adagio
hang
Silver and Gold
No. There are songs that have the word deeper in the title but not one with only that word.