Asklemmy

It's slangy, for sure. It's said in a way where the rest of the sentence is not explicitly stated but implied. Like, "Now I put the cheese... " (On the burger), and then I put the bun. (Again, on the burger.) It's not terribly uncommon, but it does happen.

No for most people because "put" would require two complements - the object and the place. (It's a lot like a ditransitive verb, even not being one.) And in this case the place is missing, you'd expect it to be at most replaced with an adverb (e.g. "now I put the cheese there").

You'd say "Now I put the cheese in" or "there" or something of the like

It's colloquial and you'll hear it when people talk about making food. Like if you're making a sandwich. You put mayo on the bread, then you put the cheese, then you add meat and lettuce or whatever.

It's kind of like "on" is implied and you don't bother to say it. I just mentioned it, so I don't need to say it again. That's how it feels to me anyway.

I could see myself saying "First you put mayo, then you put cheese." That would be like if someone was standing next to me, watching me make the sandwich. They can see exactly where I'm putting things. But normally you do want to specify where you're putting something.

As a native English speaker it sounds strange to me. I have heard it used before by a fluent English speaker who was not a native speaker and it made sense, but caught me off guard at first.

It reminds me of how my French grandma would say "shut the lights" instead of "shut off the lights", or "shut the lights off". It makes enough sense to understand, but doesn't feel quite right.

The phrase that immediately comes to mind is "poner la mesa." It translates from Spanish as "to set the table" but literally means "to put the table." Similarly, one might take photos with a camera, but the literal meaning of the Spanish phrase "sacar photos" is "to remove photos."

Linguistically and colloquially (the latter lending more to your example of adding cheese to something), we often use weird verbs in specific contexts.

What you've described might not fall into the category of proper grammar, but it also doesn't come across as strange or unexpected.

There are also unspoken rules about the ordering of adjectives to ~~blindly follow~~ follow blindly.

Edit to add: Proper, verbose grammar is also not usually necessary or even useful in the context of directions/commands, particularly in lists of such. Sure, "add cheese" would probably make a technical writer happier, but the point still gets across.

The context is important as well:

"What toppings do you put on your burgers?"

"I put cheese."

No, it's definitely not grammatically correct.

It is often heard from non-native speakers and will probably be understood, but in the absence of other context it will be perceived as slightly odd. Perhaps it's on the way to being widely recognized as fully "correct" but I don't think it's there yet.

I have heard this before, mostly from people with English as a second language. It sounds a little odd to my ear and probably isn't technically correct, but like most things in English, if it makes sense in context you can let it slide.

Plop the ham thusly, please

In golf it can be used without preposition but also without object, and with completely unrelated meaning.

Not grammatically correct, (except in extremely specific circumstances) and common in ESL.

It's widely used, so I wouldn't draw some connotation from it.

Instructions can feature a fair amount of ellipsis: Put ham. Put cheese. Put bread.

Everybody knows it's ellipsis, and they'll be a little put out if you expect full sentences instead of rapid fire and terse instructions.