"My gut tells me" This means that you have an intuition that something is right or wrong, but you don't really have an solid evidence. "Guts" are your physical insides... your stomach, blood, internal flesh. All the yucky stuff that you see in horror movies! :)

Both are correct. The singular form is from the original idiom. The plural form is the literal version. However, using as an adjective, it is always singular, "my gut feeling tells me that..."

I can't understand that you mean this phrase. It sounds a little bit strange. But the second is better in any way.