Statement (I): No year can have 5 Sundays in the month of May and 5 Thursdays in the month of June.
A year has 5 Sundays in the month of May => it can have 5 each of Sundays, Mondays and Tuesdays, or 5 each of Saturdays, Sundays and Mondays, or 5 each of Fridays, Saturdays and Sundays. Or, the last day of the Month can be Sunday, Monday or Tuesday.
Or, the 1st of June could be Monday, Tuesday or Wednesday. If the first of June were a Wednesday, June would have 5 Wednesdays and 5 Thursdays. So, statement I need not be true.
Statement (II): If Feb 14th of a certain year is a Friday, May 14th of the same year cannot be a Thursday
From Feb 14 to Mar 14, there are 28 or 29 days, 0 or 1 odd day
Mar 14 to Apr 14, there are 31 days, or 3 odd days
Apr 14 to May 14 there are 30 days or 2 odd days
So, Feb 14 to May 14, there are either 5 or 6 odd days
So, if Feb 14 is Friday, May 14 can be either Thursday or Wednesday. So, statement II need not be true.
Statement (III): If a year has 53 Sundays, it can have 5 Mondays in the month of May
Year has 53 Sundays => It is either a non-leap year that starts on Sunday, or leap year that starts on Sunday or Saturday.
Non-leap year starting on Sunday: Jan 1st = Sunday, Jan 29th = Sunday. Feb 5th is Sunday. Mar 5th is Sunday, Mar 26th is Sunday. Apr 2nd is Sunday. Apr 30th is Sunday, May 1st is Monday. May will have 5 Mondays. So, statement III can be true.