trying to build a run for them that will be comfortable enough for them until we can build a much larger one next summer, so what is the min. size you would recommend for all 12 chickens
I don't look at coop by itself and run by itself but the two together as a space system. How you manage them together helps define your needs. And you are in New York, you will have some serious winters. Not so much from the pure cold but from the snow and cold winds, that can make your run unusable for a significant portion of time. People deal with this all the time in different ways. How big is your coop? Is it big enough so the chickens can stay in it without going outside? Many times they can, even if it isn't that big. Often people block of a portion of the run to keep snow and wind out, maybe by putting tarps over it. A covered run can be nice.
I have yet to see a chicken with a tape measure making sure they have a certain number of square feet to call their own. One way chickens have learned to live in a flock is in case of a conflict the weak runs away from the strong. They need enough room to get away. And the weak avoid the strong so there is no conflict to start with. They need room to avoid entering their personal space to reduce the chances of conflict to begin with. Once they are fully integrated and the pecking order is set there is usually very little conflict, that's why so many people do well with fairly little space. But you are going to be integrating young chicks, that takes extra room. I'm glad you are doing that in the good weather months, not in winter.
You can manage this by pure room, free ranging is nice but most of us can't do that. Having separate areas can really help. Don't be too surprised if you see your chicks in the coop when the adults are in the run. When I go down in the morning to let them out I typically find my chicks on the roost while the adults are on the coop floor. They are avoiding each other and getting along great. You can improve the quality of what room you have by adding clutter. That means having things they can hide under, behind, or above (like the roosts in my coop).
How are you building the run? Will it be a fence with posts and wire? If so, you have a lot of flexibility in how you shape it. If you are building something with 2x4's or other lumber, the less expensive sizes are usually 4'or 8' in dimensions. I'd prefer an 8x16 as opposed to a 10x12 run because it is more efficient to build. If you are covering it, the 8' span could be easier to cover with standard 8' lumber and get it to support a snow load. While my personal preference would be bigger, either one should be OK for a permanent run.
how predator proof does a run need to be if chickens are only out during the day and secured in the locked coop at night? Should I still do the run as secured as if they were in it 24/7?
How devastated would you be if something happened to a chick or chicken? That may determine how predator proof you want the run.
Like Aart, mine is not predator proof. My 12 x 32 main run is predator resistant. It is also the 2x4 welded wire with aprons. I have a solid cover over about 2/3 of it and netting over the rest. A raccoon, bobcat, fox, or possum could climb into it through the netting but they haven't yet. I also have an area over 3,000 square feet inside electric netting. That will stop any ground based predator but it is vulnerable to flying predators like hawks, owls, and eagles. I lock mine in a predator proof coop at night and don't have issues then. Over the years I lost one to a hawk and one to an owl in the electric netting area.
I agree, bobcat, mink, fox, coyote, raccoon, and practically every other predator can hunt during the daylight. I even saw a possum eating at my compost pile at 1:00 PM on a bright sunny day. I did not expect that. Most of these are more dangerous at night because there is no human activity to scare them away but they are still a risk.