I am worried that you have created q2=Root[ ,2] without a detailed understanding of the numbering of Root. I would guess that by your choosing to number your Root[ ,2] you may have forced Mathematica to assume things about your root that are incompatible with you then imposing additional constraints on b. If I were going to try this I would tell Mathematica what the constraints were, look at the results and try to decide which of those was the one I wanted.
If I ignore all that and just try
q2 = Root[(48 a^2 + 16 b) #^4 - (40 a^3 + 168 a b) #^3 + (-45 a^4 +
225 a^2 b + 72 b^2) #^2 + (27 a^3 b - 162 a b^2) # + 27 b^3 &, 2];
Simplify[Reduce[q2<c<0 && 0<b<a^2 && a<0, b, Reals], q2<c<0 && 0<b<a^2 && a<0]
then I get
3 a<4 c && b > Root[-45 a^4 c^2 - 40 a^3 c^3 + 48 a^2 c^4 + (27 a^3 c +
225 a^2 c^2 - 168 a c^3 + 16 c^4) #1 + (-162 a c + 72 c^2) #1^2 + 27 #1^3 &, 1]
so, because of the b > Root, I believe it knows that b is real and there is a constraint on the value of b.
I really doubt this resolves your uncertainties about the result.
For your second question, about whether r1<a^2, I would try to give Simplify that question and provide all the domain information you have that is applicable as the second argument to Simplify. Unfortunately I can't tell exactly what that would be from your question.