Format an empty cell as a Date of Type 14-Mar-01. Now enter the simplest formula referencing an empty cell into that cell, e.g. =A2. With A2 empty, the formula evaluates to zero. As the zero is fomatted as a date, it displays 0-Jan-00, but the underlying value is not text, it's the number zero (the formulaic value of an empty cell).
Another, and perhaps more apposite, solution would be to use a more complex formula for the "dates that pull through from another sheet," e.g., instead of a formula like this for those cells:
use a formula like this:
Then the referencing range would show blank cells for the blank dates on the referenced range and a simple MIN (as you suggest) would work with the referencing range. (The MIN function ignores any empty cells, logical values, or text values; the MINA function does not.)