All Macs should be able to do both NSTC and PAL.
The Mac should use EDID information sent down the cable from the display to detect what standards the display supports and set itself to match - however I have seen some TVs issue incomplete information.
Yes you normally can tell the Mac whether to use NTSC or PAL, this is in the System Preferences -> Displays preference pane. If you have two Macs you could have Screen Sharing turned on on the Mac mini so you can remote control it to view and alter settings whilst it is having a problem.
Mac minis are not really capable of 4K so if your TV is a 4K display this might be confusing the Mac mini, 1080p should not be a problem. The 2011 model Mac mini did seem to have its own issues with HDMI and we ended up using an active Mini DisplayPort to HDMI adapter to get round this a passive adapter was not sufficient. Our problem was the screen intermittently going black.