If p n q means p => q or negation of q implies negation of p i.e. ~q => ~p.

In direct method, we assume p to be true and use it to prove that q is also true.
There is no case for p to be false in direct method.

When we talk about this, that, here, re, we are not sure about what we are talking about so ?There are 27 days in this month? is not a statement.
?February has 28 days? and ?February has 29 days? both are not statements because February may have 28 days or 29 das based on year. ?There are 32 days in this month? is a statement as it is false. We cannot have 32 days in a month.

If all component statements can be true simultaneously n ?Or? is said to be ?Exclusive Or?.

Yes, negation of statement is also a statement. If statement is true, its negation is false or if statement is false, its negation is true.

Prime number is not divisible by 2 always. Two and two makes our not five. All integers are not positive. 100 is divisible by 2 and 5 both. So, component statements of ?100 is divisible by 2 and 5? are true.

Contrapositive statement for given statement ?if p n q? is ?if not q n not p? i.e. ~q => ~p.

?History is a boring subject? is a not a statement as it all depends on reader wher he like History or not.

We know, square of a natural is always positive. For a statement connected with ?Or?, at least one of component statements to be true so it is true.

The compound statement with ?And? can be false if any of its component statements are false. All component statements can be false. In such case also, compound statement will be false.