Instead of drinking plain water, I would like to start drinking water with mint. I plan to put several leaves of mint in a 1.5 liter bottle filled with tap water during the night, and drink the water during the following day.  So I would like to know, is it safe to leave the bottle un-refrigerated during the night? 
If you really want to render fat out of solid animal fat tissue, you need to cook it low and slow. There will be an initial period with a lot more bubbling and spattering as water is more rapidly released, then eventually it'll just be slowly cooking away, slowly releasing fat.  High heat methods like searing, broiling (UK grilling), and grilling (UK barbecuing) are pretty much the opposite of this. Everything just cooks fast, tightens up, and if you tried to cook long enough to release much fat, you'd probably just burn it all.  So I wouldn't think of fatty chops or steak as a good source of fat for this. You don't want to cook that meat low and slow, generally. So you'd probably have to trim the fat off to render slowly while you cook the meat on high heat, and there are way cheaper cuts of meat that you could get way more fat from, so why bother?  As mentioned elsewhere, fat is not evaporating. You might manage to burn a bit of it onto your pan, but given the above points, it's just that you probably just aren't getting a lot out of this. 