Should a ceiling fan on a long rod move when using the pull chain?
Fans are mounted in a ball and socket type joint, both to allow them to self level and to allow a slight movement to reduce vibration being transmitted back up through the mounting hardware to the ceiling. Slight shifting when you tug on one side of a pivoted fan is likely not a concern.

Also, 2 foot downrods (of the proper type) are not at all unusual in standard fan setups.