Translation:The doctors know my grandfather well.
I considered the same thing and immediately dismissed it as being at the very least awkward. I find it easier to understand and remember the order of this sentence by slightly switching up the English translation. Literally, "The doctors know well my grandfather." Then I change it to "My grandfather is well-known to the doctors." At least it helps me with the "bien" not being at the end of the sentence.
The definite article is never superfluous in Spanish. It is often used in Spanish when it is not used in English, but that doesn't make it superfluous.
The definite article in the Spanish sentence means either (a) we are talking about specific doctors or (b) all doctors in general. Since it wouldn't make sense for doctors in general to know any one person well, that can't be the interpretation of the Spanish. Thus, it can only be a reference to specific doctors.
In short, you need the definite article in the English translation.
"well familiar" is redundant, as that's what familiar means, to know something well:
well known from long or close association: their faces will be familiar to many of you | a familiar voice. often encountered or experienced; common: the situation was all too familiar. * [predicative] (familiar with) having a good knowledge of: ensure that you are familiar with the heating controls.
You can emphasize or strengthen it by adding "very," but "well" is superfluous.
In Spanish you need "a" in front of the object, if the object is a person or people. For example: Veo a Pedro (I see Pedro); Veo a las maestras (I see the teachers); Veo a mis amigos (I see my friends); Veo a algunos medicos (I see some doctors). It can also be used if the object is an animal and the speaker holds some personal feeling toward it.
You don't use the personal "a" if the object is inanimate...Veo el libro (I see the book).
I think this was addressed above by Marcie65Brown near the top. In short, the Spanish language keeps its adverbs as close to the verb as possible.
"Los médicos conocen bien a mi abuelo." .... bien, an adverb is kept close to conocen, the verb it modifies.
If the sentence is made slightly different by eliminating the adverb bien, then the personal a follows the verb...
Los doctores conocen a mi abuelo....The doctors know my grandfather.
Does this sentence suggest they know him personally (for example, as a friend), or could it also mean something like they know him well because he's ill so they see him a lot at the hospital? It could mean either in English, so I'm just wondering if the same dual meaning is possible in Spanish
Yes, conocer can have the full range of meanings in Spanish as we have in English. In this case, they might be very familiar with his medical history, they may have seen him often and know him personally, they may know that he is a big liar, or a hypochondriac, or afraid of needles. A very common meaning of conocer in Spanish is to "know someone's tricks" they can't fool you anymore.
The reason that "knows" is incorrect is because that is the incorrect verb form for this sentence. The ones doing the "knowing" are the doctors, which means the verb "to know" will be used in the "they" form. English does not have obvious individual endings for each verb conjugation, unlike Spanish. In this case, the verb is conjugated I know, you know, he/she knows, we know, they know.
If your grandfather's case was compulsory reading for every medical doctor in the world, then you might be able to argue that "doctors" could be used generically. However, if that was really the situation being described, the verb would be something like "saber" rather than "conocer." The latter implies first-hand knowledge, which is why it's often used to talk about meeting someone for the first time.
Is there a Spanish language equivalent for grandpa or grandaddy? What I mean is... Duolingo seems to accept grandfather and grandpa for abuelo; however, grandfather is more formal than grandpa. If I needed to address grandfather is a formal way, does 'abuelo' get adjusted in any way?