Is a vehicle still drivable with a bypassed heater core?

Depends on the state if you can and how long it works.

You can get away with it in a warm climate like as a long term solution (or a permanent bypass on high mileage beaters) in a state like Florida since cold weather is rare and only lasts 1-2 days at most, but in a consistently cold state bypassing it can be less then ideal. It is an average $800-1,000 repair, so bear that in mind.

On an older high mileage vehicle like a 200k mile Toyota I'd just bypass the thing and call it good, but if it was reasonably new I may have repaired it. Over here, we just bypass them on old vehicles that aren’t worth much due to mileage (but work fine because Japanese quality), or worth more in parts. We're not going to spend $1k on a thing you almost never need when the 200k mile Camry already works otherwise.

Yes u can drive it won t hurt a thing.

Yes you can, however you won’t have a defroster anymore so if your windshield ices/fogs up then you will have no heat to clear it.

Seems like this isn't really related to the heater core. Back when I bypassed the heater core on my old Cherokee, the AC blew freezing air but the Jeep still ran fine. The real issue here is why your rig's overheating like crazy. Your problem sounds more like the cooling system isn't circulating properly. Check if the water pump's actually working—after starting the engine, feel the temperature difference between the top and bottom hoses of the radiator. If you've got an infrared thermometer, scan the hoses for temperature changes. Don't force it to run like this.