Sounds weird (for me) in English. However, it probably makes sence if doamna can be translated as "hostess".
You are probably right. We have a word in Ukrainian, "Gazda", which means something like "householder" with an emphasis on household (or estate) management. It also has a female form, "Gazdynia", which I thought might be a good translation to "doamna casei". And "hostess" looked like the closest English word by meaning.
gazda is a word in romanian as well, it means the host. gazdele - the hosts is a term used a lot in sports for the home team
It's archaic I think in most places now, but certainly would still be used on formal occasions in some settings.
This is a phrase usually associated with door to door salesmen. It's quite old fashioned but still used occasionally, especially to young girls who might open the door to them.