Is the difference between two ordinal variables ordinal? Is the difference between two ordinal survey responses still ordinal?

Clearly not, in general.  Take a 4-level pain scale for example (none, mild, moderate, severe).  Going from moderate to severe pain may be far worse than going from mild to moderate pain.  Yet they both have a difference of 1 if pain were coded 0,1,2,3.
Ordinal scale data need to be analyzed in a way that subtraction is not used.  This is discussed further here.
