I have never heard a native speaker use sentence with participle clause such as ‘Thinking about her past, she cried bitterly.’; ‘Bitten by a snake, she died.’ in their day-to-day conversation although I see in writing.
It is slightly more common in the second half of a sentence.
"I think I saw Banksy. I looked out of my window that night and saw someone painting that mural of the woman sneezing."
I've just heard on the news, " A Chinese space capsule carrying samples of moon dust has landed in ...Mongolia, ..."
There is a cliché often used by salesmen/ piece workers angling for a job. "Seeing as it's you, I'll do it for £50."
Yes. Learning to speak English natively, I frequently use participle clauses in conversation.
One famous example is the third paragraph of Martin Luther King Jr.'s I Have a Dream speech (it has been noted that "But one hundred years later" is a complete phrase, so I have added a better example below).
But one hundred years later, the Negro still is not free. One hundred years later, the life of the Negro is still sadly crippled by the manacles of segregation and the chains of discrimination. One hundred years later, the Negro lives on a lonely island of poverty in the midst of a vast ocean of material prosperity. One hundred years later, the Negro is still languished in the corners of American society and finds himself an exile in his own land. And so we've come here today to dramatize a shameful condition.
Generally it is used, as in the speech above, to place the emphasis on a different part of the sentence than it would be otherwise.
Being a single lady, no, I won't be going.
as opposed to
No, I won't be going, being I am a single lady.
Edit to address comment: Thanks for the clarification @Geoffrey, a more precise example of a leading participle clause from the same speech above would be
In the process of gaining our rightful place, we must not be guilty of wrongful deeds.