Of my two entire males, one has had a uti once: the only symptom was prolonged standing with his leg cocked, producing nothing but staring into the distance (his normal behaviour is to cock and wee quickly, often while sniffing with gusto at the surrounding vegetation, before rushing off to the next good smell). When this abnormal-for-him behaviour did not revert to normal after a couple of days, a urine sample to the vet enabled diagnosis and ab treatment.
He does also tend to get urine on his belly-fur and feathers, but this is normal for him (he is not dainty

) and is wiped off at grooming time (I also try to keep the fur and feathers trimmed short enough to be out of range

) I wonder if this (urine) is the explanation for the stickier fur you describe? I use a very dilute solution of Hibiscrub when wiping his fur.
The other has once had a slight abnormal discharge from his penis, but this was a creamy coloured thicker fluid, not pus: it didn't get on his fur, and he didn't lick himself more than usual (it was indicated to me by the other dog's sudden interest in that region). Normally neither dog has any noticeable penile discharge, they lick themselves from time to time, but not for extended periods nor obsessively.
Red staining (unless blood) seems to be related to diet: my older dog used to get red saliva stains around his mouth, and on his paws when he licked them; since changing his diet to raw breakfast and chicken-and-oats kibble dinner, the red stains have gone. I'm sure others will be able to comment further on this - my previous dogs have been either black or red, so stains, if any, were invisible.