I wouldn't chalk them up as being unprofessional just yet. It's quite possible they had some emergencies and just didn't have you on their mind when they might be understaffed and had all hands on deck dealing with stuff.
The main thing is you DID get an email asking if you were still interested. Your chances at this point are better than slim.
In closing, if you still have misgivings about the distance or especially if you still have an issue with the company emailing you a few days later than they agreed and feel they're untrustworthy then cancel the interview.
I understand it's bad enough needing a job, and having a potential employer not respond when they said they would, but having that happen on your birthday of all days....if this has soured you on that company, so be it.