We have had Finches for over 20 years. We have 16 zebra Finches and 4 Society Finches. if you are building a cage, remember they need more room to fly side ways or across than up and down. Don't know why store birds cages are built taller than wider. Finches fly sideways. If you are building a cage, you will need a way to add stuff inside, like more nest, maybe food and water dishes and branches, perches etc. I would have openings where the food is to be, side ones for reaching in for the nest or perches. You can look online to see the different custom cages and take it from there. again, make it wider than tall.
We house both types together, but some breeds don't mix. Zebra finches are the most common with the males having an Orange spot on the side of their faces while females are all Grey or a solid color. I would figure out the sexes and get enough birds to even out the pairs. Yes, even that old they can breed. Not often but still can... We went years before any of our Finches hatched any eggs, (THEY LAID A TON OF EGGS BUT NEVER SAT ON THEM) then that one summer/winter we got 12 more out of 4 batches.. They seem to breed when there is more room. We got them a much wider cage that year, 32w X 16h. HAVE SINCE GOTTEN A BIGGER cage 36h x 52w.
Like wild birds, the males build the nest trying a land a female. once she likes the nest, they become a pair. Before that, she will check out the nest, throw stuff out of it and let the male try again. I use those whole finch nests, even thought my birds have use the half nest, they never had any hatch in a half nest... Good luck, they are fun to watch and be around.
I got chickens outside and Finches inside.
sorry this is long winded... hope I helped...