4) If both cells are numbers AND they are DIFFERENT I want them
AVERAGED and then ROUNDED. However, I do NOT want to round the average
if the averaged number contains a .25 or a .75. For example, if I
average 2.00 and 2.50, I want Excel to give me a 2.25 and NOT round
this to 2.30.

It sounds like you want MROUND(). For example:
MROUND(AVERAGE(H101:H102),0.25) will round to
multiples of 0.25.