In everyday speech, they are both fine and sound perfectly natural. However, "gonna" is not grammatically correct. If you are in conversation or texting a friend, "gonna " is acceptable but if you are writing formally (a business letter, a school paper, and etc), it is better to use "going to..."

They are both correct, however a British person would probably say "going to have a shower". Americans 'take' baths and showers while Brits 'have' them

Neither. "Going to shower" "Going to take a shower" are both fine though. (for the love of all that is good and holy, don't write or type "gonna." The person who taught you that is wrong.)

Both are fine.