There once was a time in which being a runner with flat feet was a really difficult thing to navigate. But running shoes have come a long way, and now you can find plenty of name brands, styles, and manufacturers who produce shoes specifically for runners with flat feet who need additional support in their arches, and to help absorb the foot shock that running produces upon impact. People with flat feet do not have tendons along their arches that are as quick and efficient to absorb shock, so it is important to buy shoes that can help with this.
But then, like you are saying, you have to take into consideration different running conditions and temperatures. Cold weather running means there is an increased risk for ice on the ground. Especially hazardous to runners (and drivers) are patches of black ice that can't be seen - until you slip and fall! Snow is another inclement weather challenge to runners. Running in snow means you need a shoe that is thermally insulated to help protect your toes from frostbite, extremely water resistant, and have excellent traction with which to grip the ground.
So basically, you are looking for a shoe that does both: it provides great support and comfort to help lift your arches and absorb shock, but also can grip the ground and help you avoid a potential falling injury. You can check out our list of shoes that are excellent for helping runners who have fallen arches or flat feet, found here. We also have a list of trail running shoes that are great options for snow and ice because of their traction, found here. Additionally, you can search through the different running shoes of your preferred running shoe brand to find winter specific shoes. Most of the top selling brands like Asics, Mizuno, Brooks, and Saucony will have a shoe design that is catered toward running in inclement weather conditions.
However, to get the best of both worlds, you should consider investing in an insert or a pair of insoles that are specific to helping with flat feet and fallen arches (here) and sticking it in a great pair of cold weather or trail running shoes!