It's not really correct but natives often make this mistake. No one would misunderstand it.

The structure is awkward, and the tone is a bit formal. You would probably be understood, but I would recommend saying it like this: Because I'm a doctor, people often ask me for advice. OR Since I'm a doctor, people often ask me for advice.

This sounds like a perfectly normal and natural, commonly used phrase to me, so I don't see a problem with the formality or the tone. I agree that it breaks the "rule" about participle clauses that is in your original question. It's worth remembering that rule in order to avoid creating dangling participles. However, there are quite a few set expressions that are exceptions to that rule. And as others have said, native speakers understand these phrases. Such as: "Knowing Tom, he will be late." Or "Speaking generally, most participle phrases demonstrate agreement." ;-)

natives skip the part "(As I am) being a doctor..." no one sane would connect "_a_ doctor" with "people"

"Being a doctor, people often ask me for advice" is acceptable but poorly constructed. Your assertion that there are two verbs is false. The sentence has only one subject ("people") and only one verb ("ask"). "Being" is a participle acting as an adjective. "Being" is never ever a verb and cannot have a subject. The subordinate adjectival phrase "being a doctor" modifies "me". The reason the sentence is a poor one is that an adjective ought to be close to the noun that it modifies. In your sentence "me" is too far away from the adjective phrase that modifies it ("being a doctor"), and this creates a potential for misinterpretation. It can be fixed like this: "People often ask me, a doctor, for advice." "Being a doctor, I am often asked for advice." "People often ask me, being a doctor, for advice." When you place an adjective phrase next to "people", it will naturally modify "people": "Being sick so often, people often ask me for advice." Participles cannot be verbs. They never have subjects, but they can have objects: "Seeing a cat, the boy started to dance." ("Cat" is the object of the participle "seeing". The only subject of the sentence is "boy" and the only verb is "started".) Confusing explanations of the "present continuous" word pattern create the false impression that participles can be verbs. They are not. When they act as adjectives, they modify the closest available noun: "Dancing, the boy saw a cat" ("dancing" modifies "boy") "The boy, dancing, saw a cat" ("dancing" modifies "boy") "The boy saw a cat dancing" ("dancing" modifies "cat". It is the cat who dances!)