Re: runny/running nose
As an American English speaker, I would never refer to a running nose.
You say "My nose is running" but "I have a runny nose."
I don't know why - but that's how we say it.
I may hear "My nose is runny (and my head hurts and I have a sore throat, etc.)" but not "I have a running nose.
So, in short - after the verb, either one, but before the noun, only "runny."
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.