The number one thing is allow extra time. Scout routes in advance so you don't end up time crunched and stressed out. Stress leads to bad decision making, and bad decision making doesn't lead anywhere good. With kids and extreme temperatures, it tends to lead towards EMTs being involved and that's more excitement than anyone wants. Especially the EMTs.
The trailer is a very good idea. Practice with it before you put a child in it, because weight changes the handling. Make sure the trailer has a useful maximum weight, and test the handling with it if you can. Just because it claims it can pull X kg doesn't mean it's always safe or sensible to do so. Much better to find out the limits in a low stress situation than in an emergency.
Trailers make buying shelf stable staples in bulk an easy thing. This is good, since flour and rice don't go bad easily, and are hard to carry without a trailer or Extracycle arrangement. Don't get overexcited by buying in bulk. Soda may be shelf stable, but it's not very useful when you're exhausted and have hungry kids.
Keep some instant food around that the kids like. Frozen pizza, frozen potstickers, frozen perogis, granola bars and canned soup are all things that worked for my mom. Your kids will have their own favorites. When everyone is tired and hungry, cranky and screaming is not far behind. Processed food is better than trying to kill each other.
Wool is good. Silk is good. Down is good. Cotton is bad. Silly technical fabrics are usually good, but they don't last as long as the natural stuff IME. My heavy duty polyester fleece is wearing very thin after only 5 years or so. I have had silk and wool garments last for over 10 years after being bought used. For kids, this isn't a big deal. For adults? Huge deal. Avoid disposable clothing as much as you can.
Wool insulates when it's wet. This kinda matters when it's cold and rainy. Wool socks and mittens make a noticeable difference. Wool also breathes, so a thin wool sweater can be comfortable as an only layer even in pretty warm temperatures (say, 20-25 C).