Stag : Hind :: Ram : ?

A stag is a male deer; a hind is a female deer. The relationship is male to female of the same species.
A ram is a male sheep, so the female is the ewe.
Why the distractors work: a hart is also a male deer, and a heifer is a young cow — both feel animal-adjacent, and both fail the stated relationship.

OPQ : PRT :: FGH : ?

Convert to positions: O(15) P(16) Q(17) → P(16) R(18) T(20).
The shifts are +1, +2, +3.
Apply to F(6) G(7) H(8): 6+1 = 7 (G), 7+2 = 9 (I), 8+3 = 11 (K).
Answer: GIK.
Note: option GIL is there for anyone who applies +1, +2, +3 to the first two letters and then loses count on the third.

CIL : BIG :: DAY : ?

Positions: C(3) I(9) L(12) → B(2) I(9) G(7).
Shifts: −1, 0, −5.
Apply to D(4) A(1) Y(25): 4−1 = 3 (C), 1+0 = 1 (A), 25−5 = 20 (T).
Answer: CAT.
Method: that both pairs happen to produce real words (BIG, CAT) is a coincidence of the setter's design, not the relationship. Trust the positions, not the meaning.

PEN : 35 :: RAN : ?

Sum the letter positions: P(16) + E(5) + N(14) = 35 — matches.
So RAN = R(18) + A(1) + N(14) = 33.
Method: when a word maps to a number, always test the plain sum of positions first. It is correct far more often than any other rule.

444 : 21 :: 964 : ?

The digit sum of 444 is 12, which does not give 21 — so try roots.
21² = 441, and 22² = 484. So 21 is the whole-number part of √444.
For 964: 31² = 961 and 32² = 1024, so the answer is 31.
Method: if the second number is roughly the square root of the first, check the nearest squares. The numbers here are chosen to sit just above a perfect square.

7 : 49 : 56 :: 5 : 25 : 30 :: ?

Read each triple: 7, then 7² = 49, then 49 + 7 = 56.
Check against the second: 5, 5² = 25, 25 + 5 = 30. Confirmed — the rule is n, n², n² + n.
Test the options: 3, 3² = 9, 9 + 3 = 12.
Answer: 3 : 9 : 12.
Note: 4 : 16 : 64 is designed to catch anyone who reads the third term as a further squaring rather than an addition. Always verify your rule against the second given triple before testing options.