The particle は is used here to mark the topic of the sentence. In this case the topic, overall arching thing that is being talked about, of the sentence is a doctor (医者). This happens to be the same thing as the subject, which could be marked with the particle が.
I would say the reason that は is used instead of が based on the context is that a doctor is common knowledge. Also we are setting the stage (overall arching topic) that you are speaking about a doctor. Since the concept of a doctor is common knowledge it does have to be "introduced". You could probably still use が here and the meaning would not change, but if a が was used here I would assume that the topic was already set to talk about doctors.
To answer the second part of your question (used to ask questions such as). In this case the particle が is used when asking questions where the word that comes before the が is a wh- word
Who ate the food?
You are not sure who ate the food and in this case they, whoever this is, has to be introduced.
Your sentence broken down
The portion 医者はいますか translates to
Is there a doctor that exists? -> Is there a doctor?
This is paired with the portion
Creates a relative clause and adds more information to the 医者はいますか part
So the full phrase 英語の出来る医者はいますか translates to
Is there a doctor that exists who can speak English? -> Is there a doctor who can speak English?