To address your practical examples:
For the first question, you can just simplify in an additional step:
Simplify[
Sqrt[Sin[x]^6 ((a^2 + r[]^2)^2 -
a^2 (q^2 + a^2 - 2 m r[] + r[]^2) Sin[x]^2)^2]/((a^2 +
a^2 Cos[2 x] + 2 r[]^2)^2 (-a^2 q^2 + a^4 + 2 m a^2 r[] +
3 a^2 r[]^2 + 2 r[]^4 +
a^2 Cos[2 x] (q^2 + a^2 - 2 m r[] + r[]^2))),
(a^2 + r[]^2)^2 - a^2 (q^2 + a^2 - 2 m r[] + r[]^2) Sin[x]^2 > 0
];
Simplify[%, 0 <= x <= π]
(* Sin[x]^3/(2 (a^2 + a^2 Cos[2 x] + 2 r[]^2)^2) *)
For the second question, you can use FullSimplify
FullSimplify[Conjugate[a + I Cos[θ] r], Assumptions -> {a > 0, θ > 0, r > 0}]
(* a - I r Cos[θ] *)
or ComplexExpand, which is usually more efficient for this kind of simplification:
ComplexExpand[Conjugate[a + I Cos[θ] r]]
(* a - I r Cos[θ] *)
For the third question, if you want to simplify just the Conjugate expressions, you can do
expr /. c_Conjguate :> FullSimplify[c]
You could also create more sophisticated rules based on the specific form of the expression that you're trying to simplify.