circular motion

Statement (1) is wrong
Let T be the tension in the string
since T.cos(theta) = mg, where m is the mass of the bung and g is the acceleration due to gravity
But T = W at equilibrium
i.e. W.cos(theta) = mg ----------------- (1)
cos(theta) = mg/W
Thus, angle (theta) depends on m and W, and is independent of w (the angular speed)

Statement (2) is wrong.
since T.sin(theta) = mLw^2
i.e. W.sin(theta) = m.L.w^2
if angle (theta) is constant, L and w^2 are inversely proportional. L decreases with increase of w
Statement (3) is correct.

From (1), W = mg/cos(theta)
When angle (theta) increases, cos(theta) decreases, hence W increases.

Resolving the force and you'll get 3 equations governing the whole motion,
Let T = tension of string,
T = W
Tcos(theta) = mg
Tsin(theta) = mw^2r
Here, you should note that there are four variables changing which will affect each other simutaneously, namely theta, w, T and r.
When w decreases, L must increases as T and theta must not be affected (because mg is constant)
(1) is wrong.
(2) is patently wrong with the third equation.
(3) is correct. When W increases, T increases, and as mg is constant,
cos (theta) must decreases, theta will hence increase.
Also, w and r may increase correspondingly.