I would think that only your first suggestion is correct. Il bar describes only the enitre place, I think, and not the actual table in front of the bartender. Also the "bar" at which a ballerina practices (coincidentally the same English word) is definitely not "il bar" in Italian. I think it's just one of these Duolingo sentences that help us practise our vocabulary, but end up being a little odd.
I'm still not sure about this, but I think you are right. According to the following link one can read, "Since Italian ordinal numbers function as adjectives, they must agree in gender and number with the nouns they modify: primo, prima, primi, prime." http://italian.about.com/library/fare/blfare134b.htm
Because it is not necessary with the article "the" before numbers in series. "The first, the second, etc". It is correct to say just "first, second, third, etc"., unless there is a noun after the number. i.e., "The first ball", "the third prize", etc. Here, the article is referring to the noun, and not the number.
In the given sentence, there is no noun after "ninth", and therefore no article before.
La nona because it is a woman saying she's 9th at the bar. I'm pretty sure I've seen chicks at a bar before; it can happen. ;-)
But seriously... If it was a male saying this statement, it would be 'il nono'. The 9th in this statement changes to match the gender and number of the subject.
Are they talking about the kind of bar in Italy where you get coffee (coffee bar; alcohol is not necessarily served at various 'bar' establishments in European countries) and you have to have some kind of ticket or you get served in a particular order? So is the sentence saying in effect, something like 'I'm ninth in line for coffee at this coffee bar'? (Kind of like bakeries or other places in the US where you take a ticket with a number on it, and when your number is called, it's your turn.) Can an Italian native-speaker clarify where such a sentence would make sense in Italy?