Some precum comes from Cowpers glands, which you still have - they aren't removed by the prostate removal (although sometimes they no longer work due to nerve damage). There's also a condition called arousal incontinence which is where you leak urine when you get aroused, but from what you say, it doesn't sound like that (and yet another condition called climacuria which is ejaculating urine at orgasm).
To achieve better erections, and to help with healing, you should ask for a PDE5 inhibitor (e.g. Viagra or Cialis, or more likely the unbranded versions Sildenafil or Tadalafil. Tadalafil even works when you don't have an erection to improve blood supply to the area, and there's some evidence this speeds up healing, although it's not overwhelming. Typical Tadalafil dose for this is 2 x 20mg/week (which gives you two boosts a week to try for sex together with continuous background dosing), or a 5mg daily low dose which gives background dosing, but not the two boosts). You are entitled to this as a prostate cancer patient, although sometimes it's difficult to get it from the GP and you might need the urology department to tell the GP to prescribe it.
Getting an erection when doing a #2 is often because it's compressing the penile blood vessels internally, a bit like a cock ring does, or you're straining a lot which pushes up your abdominal blood pressure (which isn't good and can cause death if it bursts a major blood vessel - one reason to try to avoid getting constipated).