The song Grandpa sings a bit of when he's being moved outside into the tent is "Tenting on the Old Camp Ground," also known as "Tenting Tonight." It was written in 1863 by Walter Kittredge and was popular during the Civil War.
The wall calendar seen at 11:00 places this episode in January 1935.
The song Grandpa sings to himself as he walks to the back porch is "Independent Farmer," an 1861 Civil War era song with music by George Frederick Root and lyrics by W.W. Fosdick. Another example of Will Geer's proclivity to ad lib with songs of this era.