"Where is the transfer counter?"
Usually, but not always. They end everything in 儿, it seems, but there are cases of 儿 being used by in a non-Beijing accent context too, 哪儿 being an example.
Actually, both of these are equivalent. The only difference is that speakers use one of them more frequently than another. People in mainland China often say 哪兒, whereas people in Taiwan say 哪裡.
I read somewhere that 哪里 is generally more formal than 哪儿. In this case, should I use only 哪里 when writing something formal, or does it not matter?
哪里 is slightly more formal than 哪儿, so between these two, I'd prefer 哪里 in formal writing. For example, if I were to write "He asks me where I am from," I'd definitely go for
他问我是哪儿人 or 他问我是哪儿的人
A transfer counter is for transfer passengers. http://www.ataturkairport.com/en-EN/preflight/Pages/Transfer.aspx
I think they're using some dialect of English other than American English. Where they say "transfer," I would say "connection" (as in: "The flight to Hong Kong is delayed, I hope I make my connection to Beijing.")
This sentence in Chinese does not say anything about an airport. Could this be a train station? I think it should just say, "where is the transfer counter?"
For train stations or any such stations I'd say 转站, literally "change stops" so change stations.
转机 is a specific phrase for transit flights e.g. Singapore - Amsterdam - Dublin.
To make the question a bit trickier, we could include in the menu of characters the character for 'ma' (for yes/no question) (as an extra character that should not be used, that is). While doing this problem, i was tempted to added the 'ma' character; probably because English, like some other languages, treats what/where/when etc. questions similarly to yes/no questions.