Welcome to Chandoo Org forums.
In cell H3, you can write a formula like below.
=IF(F3="Month End",DATEVALUE("25-"&TEXT(A3,"mmm-yy")),IF(F3="Within 7 days",A3+7,"Formula for other"))
You need to provide logic for values such as "Other (See Notes)" so that you get correct results in all three cases.