I tried to replace an old round mechanical thermostat (no apparent model number) with a digital programmable one (both Honeywell). It didn't go very well.. .

"Wire to post" set up in the old thermostat:
Red to R; Yellow to Y; Green to G; White to W; Brown to O

There was also a blue wire not being used.

And in the new thermostat, I tried this:
Red to Rh; Yellow to Y; Green to G; White to W; Brown to Rc

Using the automatic (thermostat driven) setting, I got heat+fan when set to heat, but no cooling and no fan when set to cool, and neither when set to "on".

I live in a condo complex where the furnace/cooling units are inaccessible, so I cannot trace the wires. I believe the heating is electric, but I can't find any mention in the lease papers, and my landlord has yet to respond to my query (after giving the go-ahead for me to make this change).

Any help/advice would be appreciated.

Judging by the O terminal on the original, sounds as if you have a Heat Pump system, and the new stat will not work.

If this is a condo, any owner/occupant usually has access to all mechanical systems, since the condo owner owns it.

I think you really need to check what type of system you have before proceeding any further.