why is there "van" this time at the end... is it incorrect to say "az orvos egyedül" ?
'egyedül' is an adverb, rather than an adjective, which means that we must use 'van'.
Mnemonic here, possibly. If an adVerb then use Van/Vannak; if an adjective then don't need to.
"az orvos egyedül" is not correct. I don't know how the rule goes but "van" here is not used to describe something like "az orvos okos" (the doctor is smart), it refers to physically being somewhere. I've just made up this rule. :D
Because the adverb attracts a verb (that is why it is called "adverb"). An adjective can stand in place of a verb (Az orvos okos), but the adverb defines/modifies/clarifies how the verb is to be interpreted. This concept is not different from that in the English: https://en.wikipedia.org/wiki/Adverb
Actually, it can, but an even better word for that concept is 'magányos', which is an adjective, not an adverb like 'egyedül', and comes from the noun 'magány' = loneliness. The full sentence would be then: 'Az orvos magányos'.
Unfortunately it is wrong, indeed. 'That doctor is alone.' is not equivalent to 'The doctor is alone.' in English, either. The distinction can also be expressed in Hungarian:
Az orvos egyedül van. = The doctor is alone.
Az az orvos egyedül van. = That doctor is alone.
In the second pair, 'That' is a demonstrative adjective, and its function is equivalent to 'Az az' in the Hungarian version. Note that in the Hungarian sentence the first 'Az' is a demonstrative pronoun, whilst the second 'az' is a definite article. To express the English demonstrative adjective, we need to use both 'az'. Similar constructs are quite common in German as well.