Yes, the inverter drive units keep up with cold temps. Most are rated to work down to 0F such as
THIS Gree U-Match 24kbtu UNIT which replaced my girlfriend's 30 year old gas furnace (with a constant burn pilot light!) and air conditioner. We chose this one since it could be hooked up to ductwork as designed; she didn't want a more efficient unit hacked to mate with the ductwork. That would have worked ok in her case-no return ducts and the supply ducts are short and unrestrictive. ecomfort.com was great to buy from.
It works beautifully! Her 2BR condo is reasonably efficient, and this unit runs less than half the time unless it is extremely cold even though it can throttle down to 8kbtu. To put it through its paces, I set the thermostat to max on "Turbo" mode when it was 20F outside. I burned my hand on the vapor pipe at the indoor unit, the discharge air was toasty warm, and it made the whole place uncomfortably hot in less than half an hour. At sensible settings, it used less than $20 extra electricity during the coldest months.
Conventional ones don't pump much heat below 40F which is when you need it most. They rely on resistive heat or gas to make up the difference-expensive.
A ground source unit is very efficient because you are pulling heat from a 40F or better heat source, but you need a loop field, which needs to be large because your heating dominated climate will pull its temp lower and lower in a couple years. Good field design becomes evident after the 3rd year. Putting the field on more than one side of the house opens up the area you can draw heat from significantly. Check Energy Star rated units if you want a ready made one instead of rolling your own a la "Manifesto", or to get an idea of efficiency vs airsource. There are few site locations where I think an open loop GSHP makes sense. Their ratings don't include the energy required to "pump and dump" the water, which can be very high.
Of course, you could combine ground source and seasonal storage as I describe in my thread for "fire your energy companies" efficiency, but that is a multipart harmonized system.
Cheap heat with AC capability in multiple areas? An inverter multi-split (multiple indoor cassettes with one outdoor unit). Get one heavy on the HSPF rating and lowest temp capability. Some are optimized for cooling-dominated climates. My favorite vendors I might buy from are heatandcool.com and highseer.com.
Need a ducted system? One of the "concealed duct" models like our Gree.
There is some concern about ASHP efficiency in cold wet conditions because of the need to constantly defrost. I have never, ever seen my girlfriends unit have any frost on the fins, ever, even when it was 100% humidity in the low 30's, when in the 20's, or during the one instance we had snow. These observations were made while the unit was actively heating the condo. At low temps, there just isn't that much moisture available in the air to cause frost with.
After seeing many systems and installing a few, I would heartily recommend inverter drive heat pumps of any type, and would never recommend a conventional. You will need additional heat sources to keep up if temps are below 0F, but unless you need to rely on them frequently a wood stove or portable heaters would do it.
Warm floors are awesome if you feel like doing it. I will never go back to forced air heat if I can help it.