Hello,
can there be a situation whereby the single ended outputs of a fully differential HPF will have zero DC level but when you subtract the two outputs( taken differentially), you get the output common mode level to be zero initially and decreasing to settle at a lower value after some time for the transient response?

I agree with Jerry about common mode (the common mode is set by a feedback circuit whose gain is not infinite), but the question doesn't make sense to me... if you subtract the two outputs you get the differential output. To get the common mode you average the two signals (add them together and divide by two).