Is "(it) looks like rain" the most common way of saying you think it's gonna rain soon after you've just looked up to the sky?

I think I'd say, "(I) think it's gonna rain."

I'd probably say "looks like it'll rain" or "looks like its gonna (going to) rain" rather than "looks like rain"

Yep, that has become so idiomatic it's almost ritualistic. Many of us probably don't even try to form a new sentence for that idea--"looks like rain" just comes out of our mouths. The only reason we might not say that is if the weather was just part of some larger "story". "Well, looks like it's gonna rain, so I'd better get these tools back into the shed." But if you were just expressing the simple idea that the weather patterns indicated impending rain, then you' say "looks like rain". If two buddies were just hanging out at a campsite or on the back deck, and the clouds indicated rain soon, and there was a lull in the conversation, you'd probably be surprised if someone didn't say "looks like rain".

Also common, at least in the Midwest, is “think it’s gonna rain?” The proper response is grunt or say “well…” and not follow it up with anything

Yes, in your scenario.

It's an acceptable form although there are so many different ways to say this I think it's hard to name one as "most common".

I would say 'looks like it's going to rain' or 'looks like it's gonna rain.' I haven't heard people say it without the 'going to' or 'gonna' very often ('it looks like rain'). I might also say 'Looks like it's gonna be rainy today' or something like that. I don't think there's a most common, but this type of phrasing is plenty common and you'll hear it almost everywhere (at least in the US)

There are parts of Britain where you can confidently look at the sky and say, "It looks like it's going to rain." The only occasion when this claim is inaccurate is when it is actually raining.

Definitely common in the UK. Also common are: (I) think it's going to rain It's going to rain (Look like) it's going to piss it down (this is not particularly rude in the context of imminent rain, but is definitely informal) 'going to' is often pronounced 'gunna' (and is equivalent to the US 'gonna')

It's acceptable. Personally I tend to say "smells like rain's coming", because I consider the smell of rain to be a more reliable indicator than the appearance of the clouds.

Yea, but pretty much only in the exact scenario you described.

Looks like rain. It's gonna rain. Storm coming. Sure looks black. Storm clouds. We are about to get wet. Rain brewing. Storm brewing. Rain's coming. Storm's coming. Thunder clouds.

"(It) looks like rain" would be something you'd say when describing the visual concept of rain itself. If I showed you a picture of a distant city with clouds above and rain falling, this sentence would be appropriate, though I will admit I don't often hear people discussing the visuals of rain. To answer your question, no. You need to include something that will indicate a possible event in the future: "It looks like *it is/it's going to* rain." "It looks like *it will/it'll* rain."

That phrase is both common and has been around for awhile. The 1990 pinball game *Whirlwind* plays a voice recording of a woman saying, "Uh, oh, looks like rain!" at the start of a game.

I say, "Looks like rain". Sometimes, I'll say it in a gruff voice like the main character from The Witcher (video game). It's a pretty normal way to say, "I think it's going to rain."

yup. especially if the clouds are grey, and the other person/people saw me looking up.

Sure that's common, you can also just actually say "I think it might rain" 

"It looks like rain" is the opening line of the Beach Boys' song 'Pitter Patter.' It sounds normal to me. What's more or less common, I couldn't say.

I always say “yargh! Thar be a storm a brewing’!” Like a pirate would

In addition to that, in my area we also say "(it) smells like it's gonna rain"

I live on the west coast and I would say “it looks like it’s gonna rain”

I don't know if it's the most common way, but it's certainly a common way. There's nothing wrong with it, if that's what you're asking.

Sure yes. Or ‘the weather’s about to turn’

I say this frequently. Not sure about it being the *most* common way (maybe something like "(it) looks like it's going to rain" would still be used more) but it's definitely something you'll hear frequently.

In movies I think that's common. I'd say "it looks like it's gonna rain" People are usually aware of the weather outside so I've never said that to someone, though.

that or "(it) looks like it's going to rain"

For me it would be "looks like it's gonna rain".  "Gonna" is improper, but it's common enough. 

A storms a comin'. I can feel it in m' bones. It is a little cliche to say the phrase you gave, but it is totally fine to say.