Short answer, call a tech qualified on high efficiency gas furnaces to diagnose and repair.
The leak is probably coming from the connection to the condenser, the final heat exchanger that cools the exhaust so that water is changed from vapor to liquid. Need to eliminate the cooling coil and humidifier if there is one as a source, but it is almost certainly coming from the furnace condenser.
Re the icicles, pretty typical for this type of furnace if the conditions are right, I won’t say it is OK, because it could be a problem, but as far as I know it hasn’t been solved yet.
FYI if either the combustion intake or exhaust are blocked the furnace has safety sensors that will prevent it from working.