"He cut himself making dinner."
"He cut himself while making dinner."
In the first sentence, the absence of "while" implies the act of making dinner was connected to or caused the accident. The two sentences are almost identical in meaning, but I'd use the first one when I'm annoyed--for example, maybe I didn't want to make dinner, and then I cut myself too!
"Making dinner, he cut himself."
The meaning is technically the same as above, but native speakers would never say this sentence. However, they would say
"Seeing the child in the street, the man ran to save him."
I think whether the -ing clause sounds natural at the beginning is case by case. I don't know of a rule.
With "get," I can't think of a sentence that sounds natural with "getting" at the beginning.
"I broke my toe getting into the shower."
"Getting into the shower, I broke my toe."
The first sentence is okay. The second is awkward.