Ceiling Fan wiring

[jdvmax]

Up front I appreciate the help. I live in a house built about 1922. I removed a ceiling fan today and when I took it down there are only 2 black wires. Now I have read in other appends about determining which is hot. I used a tester that you just hold next to the wires and one does beep like it is hot and the other does not so I suspect it is the neutral. Three questions: One, when I removed the fan I believe the white(neutral) wire on the fan was connected to the ground wire on the fan bracket. Does that make sense? Two: When I used the tester like I mentioned, with the wall switch on, the tester does beep rapidly for the one black wire and not the other, but when I turn off the switch, the tester still beeps much slower for the same black wire that beeped rapidly before when the switch was on. The other wire does not beep at all no matter what position the wall switch is in. Doe it make sense that the one black wire beeps at all when the wall switch is off? Three, when I wire in the new fan, should I wire the one black wire that appears to be the neutral to the white on the new fan and the black that appears to be the hot to the black on the new fan? I would assume so. As I understand is the case with old wiring I do not see any ground wire attached to the box in the ceiling, so when the old fan's white was connected to the green wire on the fan bracket, I guess it really was not connected ultimately? Thank you again for any help.