First, let's see what the expressions might mean:
"Bater à porta" always means "to knock on the door".
"Bater a porta" almost always means "to slam the door". In some very unusual and specific contexts, it may also mean "to collide/crash/hit the door onto something or someone".
"Bater na porta" usually also means "to knock on the door". However, accordingly to the context, this may also mean "to collide/crash into the door" or "to hit the door".
"Bater com a porta" may mean "to slam the door". However accordingly to the context, it may also mean to "to crash into the door", "to collide with the door", "to smash into the door" or "to come and face the [closed] door". In some very unusual and specific contexts, it may also mean "to beat someone using a door".
To make it clear, let's see some examples:
Examples:
[en] Somebody knocked the door this morning. When I opened it, it was my mother.
[pt] Alguém bateu à porta hoje de manhã. Quando abri, era a minha mãe.
[pt] Alguém bateu na porta hoje de manhã. Quando abri, era a minha mãe.
These two are ok and they have the same meaning as the English phrase, "bateu" is "knocked".
[pt] Alguém bateu a porta hoje de manhã. Quando abri, era a minha mãe.
This is WRONG, "bateu" is "slammed".
[pt] Alguém bateu com a porta hoje de manhã. Quando abri, era a minha mãe.
This is WRONG, "bateu com a porta" may be "slammed the door", "came and faced the closed door" or "collided with the door".
[en] If nobody answer to the doorbell, knock the door.
[pt] Se ninguém atender a campainha, bata à porta.
[pt] Se ninguém atender a campainha, bata na porta.
These two are ok and they have the same meaning as the English phrase, "bata" is "knock".
[pt] Se ninguém atender a campainha, bata a porta.
[pt] Se ninguém atender a campainha, bata com a porta.
These two are WRONG, "bata" is "shut" or "close".
[en] Don't let the wind slam the door.
[pt] Não deixe o vento bater a porta.
[pt] Não deixe o vento bater com a porta.
These two are ok and they have the same meaning as the English phrase, "bater" is "[to] slam".
[pt] Não deixe o vento bater à porta.
This is WRONG, "bater" is "[to] knock".
[pt] Não deixe o vento bater na porta.
This is WRONG, "bater" is "[to] knock" or "[to] hit".
[en] Boy, don't slam the door like that! This makes a lot of noise!
[pt] Menino, não bata a porta desse jeito! Isso faz muito barulho!
[pt] Menino, não bata com a porta desse jeito! Isso faz muito barulho!
These two are ok and they have the same meaning as the English phrase, "bata" is "slam".
[pt] Menino, não bata à porta desse jeito! Isso faz muito barulho!
This is WRONG, "bata" is "knock".
[pt] Menino, não bata na porta desse jeito! Isso faz muito barulho!
This is WRONG, "bata" is "knock" or "hit".
[en] Ouch, I hitted my foot in the door!
[pt] Ai, eu bati o meu pé na porta!
This is ok and the meaning is the same as the English phrase, "bati" is "hitted".
[pt] Ai, eu bati o meu pé à porta!
This is WRONG, the phrase does not makes sense anymore, and it would be translated back to the English (needing some adaptations to be translateable) as "Ouch, [something] hitted my foot [when I was] at the door".
[pt] Ai, eu bati o meu pé com a porta!
This MAY BE WRONG. It admits the original English phrase translation, but it also admits some different ones like "Ouch, I hitted my foot using a door".
[pt] Ai, eu bati o meu pé a porta!
This is WRONG. The phrases is not well-formed and do not makes any sense at all.
[en] My neighbour is cruel, he always hits the door in the dog.
[pt] Meu vizinho é cruel, ele sempre bate a porta no cachorro.
[pt] Meu vizinho é cruel, ele sempre bate com a porta no cachorro.
These two are ok (the phrase translation, not the act of being cruel to animals) and the meaning is the same as the English phrase, "bate" is "hits".
[pt] Meu vizinho é cruel, ele sempre bate à porta no cachorro.
This is WRONG and it is hard to make any sense of it. This translates back to the English as "he always knocks on the door on the dog" which is close to "he always rings the doorbell on the dog" if you prefer to ring the doorbell instead of knocking the door.
[pt] Meu vizinho é cruel, ele sempre bate na porta no cachorro.
This is WRONG. The phrases is not well-formed and do not makes any sense at all.
[en] The car crashed into the garage's door.
[pt] O carro bateu na porta da garagem.
This is ok and the meaning is the same as the English phrase, "bateu" is "crashed".
[pt] O carro bateu com a porta da garagem.
This is very similar, but a bit different. It translates back to English as "The car crashed with the garage's door."
[pt] O carro bateu a porta da garagem.
This is WRONG, because now the meaning changed to "The car smashed the garage's door". It is similar but it is different, now the thing that was probably damaged or destroyed is the door, not the car. It could also be translated into the bizarre phrase "The car slammed the garage's door."
[pt] O carro bateu à porta da garagem.
This is WRONG, "bateu" is "knocked".
[en] Ok, funny boy, it is time for a few more torture! So, today I will hit your arms with the door.
[pt] Ok, engraçadinho, é hora de mais um pouco de tortura! Então, hoje eu vou bater com a porta nos seus braços.
This is ok (the phrase translation, not the act of torturing someone) and the meaning is the same as the English phrase, "bater" is "hit".
[pt] Ok, engraçadinho, é hora de mais um pouco de tortura! Então, hoje eu vou bater a porta nos seus braços.
This is very similar, but a bit different. It translates back to English as "I will hit the door into your arms."
[pt] Ok, engraçadinho, é hora de mais um pouco de tortura! Então, hoje eu vou bater na porta nos seus braços.
[pt] Ok, engraçadinho, é hora de mais um pouco de tortura! Então, hoje eu vou bater à porta nos seus braços.
This is WRONG, "bater" is "knock". So it would come back to English as something like "I will knock on the door [being] on your arms". At least, this time this sounds romantic instead of cruel and violent.
[en] I will slam the door in his face if he comes again.
[pt] Eu vou bater a porta na cara dele se ele aparecer de novo.
[pt] Eu vou bater com a porta na cara dele se ele aparecer de novo.
These two are ok and they have the same meaning as the English phrase, "bater" is "[to] slam".
[pt] Eu vou bater à porta na cara dele se ele aparecer de novo.
[pt] Eu vou bater na porta na cara dele se ele aparecer de novo.
These two are WRONG, "bater" is "[to] knock".
[en] I din't saw that there was a glass door when I were running, I collided with the door in my leg.
[pt] Não vi que havia uma porta de vidro quando estava correndo, bati com a porta na perna.
This is ok and is has the same meaning as the English phrase, "bati" is "[I] collided".
[pt] Não vi que havia uma porta de vidro quando estava correndo, bati na porta na perna.
This is very similar, but a bit different. It translates back to English as "I collided in the door in my leg."
[pt] Não vi que havia uma porta de vidro quando estava correndo, bati a porta na perna.
This is very similar, but a bit different. It translates back to English as "I collided the door in my leg."
[pt] Não vi que havia uma porta de vidro quando estava correndo, bati à porta na perna.
This is WRONG, and they don't makes any sense at all. It would translate to "I knocked in the door in the leg." which does not makes sense.
Well, that was a long list of examples. So, it is time to actually answer your questions:
1. In bater a porta, does the a correspond to at or to the?
Roughly, the "a" corresponds to "the" and the "à" corresponds to "at" or "at the".
Or is that form just a typo?
It is not a typo.
2. Is the choice from among à/na/a related to a different meaning or dialect?
It has nothing to do with dialects, it is all about the possibly wildly different meanings.
3. Can I take something from this example that will help me to understand prepositions in other contexts (e.g., adjacent to other verbs)?
The verb "bater" is particularly a very troublesome one. As you can see from the examples, it can have a lot of different meanings accordingly to the context: "to crash", "to hit", "to smash", "to collide", "to knock", "to slam", "to beat". However when aplied to physical objects, it always means some form of something colliding with some other thing. But, to make thinks worse, it sometimes might be used as something completely different like "to correlate" or "to match", as in the phrase "Percebi que o número de telefone do usuário bate com o número que foi bloqueado" which translates as "I perceived that the user's phone number matches with the number that was blocked".
So, I think that the verb "bater" is a very bad choice to help you understand prepositions in other contexts. It is better to start with some other verb that does not features a lot of differently context-dependent meanings.
For example, does [verb] + com + [object] usually suggest a more vigorous manipulation of the object?
No, this has absolutely nothing to do with vigour. It really means just "with". However, sometimes to make sense or just because it would be reduntant, in the translated English the "with" might disappear. Like in those examples:
"Ferrar com o plano" -> "To screw with the plan" -> "To screw the plan" -> "Ferrar o plano"
"Plantar com as sementes" -> "To plant with the seeds" -> "To plant the seeds" -> "Plantar as sementes"
"Acelerar com o carro" -> "To acelerate with the car" -> "To acelerate the car" -> "Acelerar o carro"
"Bater com a porta" -> "To slam with the door" -> "To slam the door" -> "Bater a porta"
Or, the "with" might be transformed into something else to make the phrase sound better (or kept as is):
"Roubar com uma faca" -> "To rob with a knife" -> "To rob using a knife" -> "Roubar usando uma faca"
"Dirigir com o cinto de segurança" -> "To drive with the seatbelt" -> "To drive using the seatbelt" -> "Dirigir usando o cinto de segurança"
"Cavar com uma pá" -> "To dig with a shovel" -> "To dig using a shovel" -> "Cavar usando uma pá"
In other cases, the "with" should be kept as it is (removing it would change the meaning of the phrases):
"Viajar com minha esposa" -> "To travel with my wife"
"Andar com um amigo" -> "To walk with a friend"
"Cantar com a plateia" -> "To sing with the audience"
"Desenhar com as crianças" -> "To draw with the kids"
Note that all the cases where the "with" should be kept, disappear or be replaced by something else happens after the translation and happens only to make the phrases not sounding weird in English, so this is just an artifact from the translation process.
