Your first problem was you weren't using your compare symbols correctly.
< less than
> greater than
<= less than or equal to
>= greater than or equal to
To answer your other questions; get the condition to work on every cell in the column and what about blanks?
What about blanks?
Add an extra IF
condition to check if the cell is blank or not, if it isn't blank perform the check. =IF(B2="","",B2<=TODAY())
Condition on every cell in column