Certainly. There is nothing about the recording process that makes a song stick with you. There is little doubt that someone left the Theater an der Wien in 1808 humming the opening bars to Beethoven's Fifth Symphony. Now, finding someone to play the tune that was stuck there was likely much more difficult.

Sure. We have literary references for that kind of thing.

In 1845 Edgar Allan Poe wrote (in The Imp of the Perverse):

It is quite a common thing to be thus annoyed with the ringing in our ears, or rather in our memories, of the burthen of some ordinary song, or some unimpressive snatches from an opera. Nor will we be the less tormented if the song in itself be good, or the opera air meritorious.

Mark Twain wrote A Literary Nightmare in 1876. It's a story about an encounter with this contagious rhyme that occupies his mind for days until he manages to get rid of it by passing it to somebody else. Using this as an example might be stretching things a little, but it kind of describes the idea of "an earworm."

There's this Beethoven trio with the nickname of "Gassenhauer." That word means something close to street song. See, the third movement of that trio is a set of variations on a theme. The theme is a silly tune (Pria ch'io l'impegno, "before I go to work") from an opera presented in Vienna in the last years of the 18th century.

Beethoven used that silly tune as the base for this movement because the damn thing was VERY popular. It was catchy enough to be heard in the streets of Vienna, I guess it's reasonable to say it stuck in people's heads if you could hear people singing/humming/whistling it in the street.

You can listen to that trio here (the silly tune is heard at 14:16, when the 3rd movement starts).

Composers of that period very fond of catchy melodies.