Biking in the winter more difficult? I have two kids (about 70lbs/30kgs combined) whom I during the summer frequently took biking in a trailer (once I got them used to it, say a dozen trips this summer or slightly more).  I biked 5+ miles round trip with them without significant difficulty; I'm not in great shape, but I was generally able to get up over 10MPH when I had a decent flat straightaway, and didn't need to stop (biking 3 miles in a residential setting with frequent stop signs/lights in 20 to 25 minutes).  I typically bike in the 2nd forward gear, 4-7 rear gear (4 is for starting from a stop usually, then I go up to 7 once I'm at speed assuming I don't immediately stop again).

I hadn't biked for a couple of months perhaps, when I suddenly had a need to bike ~6 mi round trip to take my older son to preschool and younger to daycare.  This was on a suddenly cold day, below freezing - around 10˚F/-10˚C.  I checked my tires visually and by feel, and they seemed fine (the bike was stored in an unheated garage, but it probably did have a 20 or more ˚F buffer from the outside given this was the first day of real cold weather).  

I made it half a mile, in 2nd and 1st forward gear, completely out of breath and almost vomiting.

I saw at least two problems: one is that it was as if I had shifted to 3rd forward gear, as far as how hard it was to use the gears - I couldn't go at all in my normal gear range.  Second was that the cold air seemed to have significant effect on my lungs even after not very much exertion.  

Are some of these issues things I can do something about, or is biking in the fairly cold impossible for someone of my skill level?  Is the chain oil wrong and not functional at lower temperatures?  Is the 70lb weight on the back of my bike making a bigger difference (due to air pressure in the tires or something)?  I'm not taking into account ice or snow here - this was the first cold day of the winter, so just simply cold (this was during the 'polar plunge' or whatever silly name they gave it this year).  I know exertion in the cold is a different beast than exertion in the summer, but this seemed a level of magnitude harder than I anticipated.  

I have a hybrid with factory tires, ~$400 USD bike when new (now 2-3 years old), probably has maybe 200 miles on it total.  No particular maintenance done recently, just usual tire airing.
I bike all winter long, down to -30°C, there are some definite tolls to your bike at colder temperatures. Unlike a vehicle that warms up after the engine has been running for a while, bikes stay cold when you ride them cold. There are two things I notice the most, air pressure is definitely one of them; cold air shrinks, and low tires drag, so if you're riding dry roads I would certainly check to your pressure. Normally I don't put extra air in my tires, because up here the cold is usually accompanied by snow, and low tires give you better traction. The other thing I notice when it gets really cold is the grease in my bearings gets thicker, and as a result my hubs don't roll as fast. Pick your bike up and give the wheels a spin, you might find that they slow to a stop a lot quicker. I notice it in my headset too, but that doesn't affect speed.

One thing I've done to avoid drag due to thick bearing grease is to actually replace the grease in my wheels with Phil Wood Tenacious Oil. I tried it out as an experiment one year, and I've never had problems with it since (I kept checking inside the hubs for a while expecting to find them dry and rusty, but Phil's oil lives up to it's name; it's tenacious).  

Of course the best thing to do is to not push yourself, compensate for the conditions, leave a bit earlier, find a new gear range that you can maintain a comfortable cadence in without giving yourself a heart attack, and just plough along. If the conditions are miserable already, there's no sense in making yourself more miserable.