The functions Rs and Rp become complex for \[Theta]0 = ArcSin[50/69]. You can insert that value into the integration interval as {\[Theta], 0, \[Theta]0, \[Pi]/2}, which silences the messages. You are using the symbol R with two different meanings, which may not be a good idea. The. I am afraid NIntegrate does not have a special notation.

I am not an expert in integration. In your case for the integrals without Abs I could find an analytical primitive, but it was not easy to use, because it has a discontinuity in the imaginary part. Still, I think you can find an exact result in that case, with some patience. If you are content with a floating-point result, NIntegrate can be more than enough. I think that your warnings come because the algorithm somehow senses the singular point, but you have to study every case on its own.