Why is the new high efficiency furnace blowing cold air sometimes

Can you look at the furnace during the time it is blowing cold air and see if the flame is on?

The thermostat has the ability to control the fan, but typically the thermostat sends a "turn heat on" signal to the furnace and the furnace controls are setup to turn the flame on for a X amount of time (to warm up the air) and then turn on the fan once the timer expired (so the air coming out of the vents is not cold). When the heat is turned on is the flame and fan going on at the same time?

Not sure about your furnace, but some have a couple of LEDs that are lit up and can be used to read a status or error codes. If you have the manual you could try to decipher the code while the cold air is running and see if it gives you a clue.

I would start by trying to figure out if the problem is with the thermostat or the furnace by installing a new thermostat and seeing if the problem goes away. If the problem is still there you can just return it. I had a furnace issue where it was going off randomly and would work in heat mode but not cool mode. It turned out that the thermostat was bad and sending intermittent signals.

It definitely sounds like that section of duct has a leak or travels through an un-insulated part of the structure. Sometimes the cavities between walls or under floor joists are used as duct or returns and they are not enclosed. Leaks in the building envelope, especially with such cold weather, could keep that area cold all the time and any warm air blowing through it will get cooled down right away.

If you are feeling a draft out of that vent when the heat is not blowing, that is concerning and may indicate a large air leak. There are a variety of methods for testing for duct leaks. One common method is by using a blower door on an external door to get a baseline leakage number for the home. Then, all of the vents are covered and another leakage number is obtained. The difference indicates the leakage of just the duct system. A service that does this is usually inexpensive (under $100) and can tell you just how much air the ducts are leaking in additional to helping you find other sources of cold air leaks into your home.

If the landlord isn't willing to pay for investigating it, you may want to pay for it yourself since you could then bring him/her hard numbers and ultimately save on your heating bill once repairs are made. If the landlord still will not budge, I'd make the claim that warm building air mixing with cold duct or external air could cause mold growth and make you sick.