It’s a fine sentence but not too clear by itself. We usually say ‘a little short’ meaning that we have most of the amount but not all. Saying that you’re ’very’ or ‘really’ short is vague. Clearer would be something like ‘I can’t pay my rent’ or ‘I don’t have money for rent’. Use ‘really’ to contrast with ‘a little’. Last month I was a little short and my parents helped me out after giving me a long lecture. This month I’m really short and afraid to tell them.

Yes, your sentence is fine and clear. It conveys that you are short on money this month. For added clarity, you might consider adding a bit more context, like this: "I'm really short on money this month." However, your original sentence is understandable as it is.

Yes, that makes sense, as long as the context has made clear that it is indeed finances that you're speaking about. What the sentence implies is that the speaker has not made enough money to pay for the month's usual expenses. Maybe it is rent, or a mortgage payment, or some other basic necessity—this would be the assumption.