These are pretty advanced questions, so congratulations — good work so far! 1. You could also use “the” to refer to a definite squirrel. If you’d like to be more emphatic about it, you can say “that squirrel.” (“The” was originally a reduced form of “that”) 2. In most cases, it would be fine to say “the bank” or “a bank.” “The bank” suggests you have a specific bank in mind (the bank where your friend has an account, the nearest bank, the only bank in your small town, whatever). “A bank” suggests that any bank would do. In practice, either article would work. 3. It’s “the” doctor because you’re referring to the doctor who normally treats the person you’re talking to (or who eats the apples). In American English at least, we refer to our doctor as “the doctor.” If we say “a doctor,” it means we don’t have any doctor in mind. Keeping healthy will not prevent you from accidentally bumping into random doctors in the street, but it will help you to avoid going to see the doctor who usually takes care of you. It’s also possible to use the possessive (which usually implies definiteness) — “my doctor.”

You're talking about different uses of the definite article. I mean that 'The squirrel is a nervous animal.' is not using 'the squirrel' in the same way as 'The squirrel in the tree over there.' In the first sentence 'the squirrel' means, as you pointed out, the whole category of 'squirrel', the whole species or population of squirrels as a group. In my second example 'the squirrel in the tree over there', 'the squirrel' refers to only that one specific squirrel. So yes, to refer to an individual squirrel you also use 'the'. This use would come under what you call Rule 3: an object that has already been mentioned. So previous to saying The squirrel in the tree over there' you would have had to previously mentioned the squirrel. For example 'A squirrel just dropped an acorn on my head. The squirrel in the tree over there.' 'Obvious' is subjective, but still a strong adjective that means 'almost objective' in a way. And yes, in that example you could also say 'a bank'. The way it is used there, 'the bank' is again referring to the category of banks. All banks/any bank. A saying like 'An apple a day keeps the doctor away' is something you say with the understanding that most people would only be examined or treated by ONE doctor, or at least one at a time.

Simplified in short everyday sentences: 1) The/A red squirrel. "The/A red squirrel over there under the tree" 2) your sentence is correct because 99% of listeners of your sentence would know you mean any bank or their bank or even your bank, but in the case of you ever meeting a pedantic person who insists upon clarity you could say specifically. "You can pay at my bank" "you can pay at any bank" "you can pay in that/this bank" "you can pay in your bank" "you can pay in (A/ANY) high street Bank" "you can pay in the high street bank" indicates you mean any high street or this high that you are on or the high street where the person lives or is visiting or next comes across etc. "you can pay in/a/at a Barclays bank" "you can pay in/a/at a /in any state owned bank" "you can pay in/a/at a/in any private bank" 3) it is an idiom so it does not have to be entirely logical, but it is because it refers to the person's​ own doctor or any doctor they may see or have in the future. British and American English we say "our Doctor" "my doctor" 'your doctor" their doctor" the School Doctor ETC. It's always singular.