When you have coolant leaking in from this spot, this is the sign that the Heater Core is faulty and will need to be replaced. Unfortunately, this is a HUGE job. The Heater Core sells for around $144, and the labor time guide calls for 6.1 hours to complete the job. You best bet here is to find a shop that specializes in HVAC Systems and are experienced in Heater Core replacement. Hopefully they can cut some of that labor time off seeing as they are experienced.
Yes. Try not to drive too long, too far. Just keep an eye on the Coolant level and you should be fine. If the coolant system empties, you run the risk of over heating the engine, and causing permanent damage.
Unfortunately not. This is a classic sign of a faulty heater core. You'll notice if you turn on your heat, it'll be luke-warm at best and/or it will sometimes present with "steam" coming from the dash vents with the heat on.