Yes, they are all more or less interchangeable. You're probably more likely to hear/say that 'the apartment is available now,' as it sounds more positive/optimistic. At least in the US, anyway. For advertising and marketing purposes, companies often avoid phrasing things in a way that would make the product sound undesirable--to say it's vacant or empty draws attention to the fact that no one is living there, and the connotation could be that maybe there is something wrong or undesirable about the apartment.

If it is clear from the context that you are talking about vacancy / availability, you could use 'empty'. However, empty could also mean that there is no furniture in the apartment, or if you were 'casing the joint' that the occupant is not there at the moment...

None of the three sentences has a unique exact precise meaning. All must be interpreted in context. Even the third one, which comes closest to actually saying the apartment is available for rent, could mean other things. For example, it could mean it is available for inspection though you cannot rent it because someone is living there.

Vacant or available yes, empty no

Not interchangeable