I do not know what that brown stuff is on your fruit tree limbs. Looks kind of like dog poo in this slightly out of focus image. I see moss which is quite natural and not a problem. I see some orange things but they are out of focus and so I can not determine what that might be. The bottom part of the brown stuff looks kind of like a liverwort. Have a look here and see if this discussion makes sense for you:

https://pnwhandbooks.org/plantdisease/pathogen-articles/nonpathogenic-phenomena/living-land-mosses

As for when to prune, this is a good time and there is no need to wait until night. Cuts now will help new buds and shoots develop. Cut during the summer will take some vigor away from the tree. You might see various organisms colonize the fluid flow but will not be a concern. Not sure why it is dripping in the first place. 

The main "trick" is just to not leave a stub but prune back to a branch or bud. Also, to not cut into what is called a branch collar which is just that swelling at base of a branch near the trunk. You have made some heavy duty cuts and it will take awhile to heal.

Turns out the brown stuff that looks like poo is kind of what I thought. This is from our moss/lichen expert:
"that it is the lichen Melanelixia glabratula. It’s usually flat, appressed to the bark, so it must be overgrowing something lumpy. Btw, you can see the grainy texture on the surface, which are minute asexual propagules containing both the fungus and the alga."