I would double-check the temperature under the heat lamp, and consider whether to put it lower down. The way it is now, it is making a large area warm, but I don't know if any of it will be warm enough during the coldest part of the night. If you put the heat lamp lower, it heats a smaller area, but it makes that area warmer. This is helpful during the cold part of the night (warmer), but also helpful in the daytime (warm area is smaller, so there is more room for the chicks to get away from the heat.) I would aim for at least 90 degrees under the heat lamp at the coldest part of the night, and it could be quite a bit warmer during the daytime.
You don't want to adjust the heat lamp several times each day: just give the chicks lots of cool space and one warm space, and they will handle the adjustments by moving themselves to whichever spot is comfortable at the time.
Does the heat lamp have a way to hang it from a chain, rather than using the clamp? Sometimes clamps get knocked loose, and of course a fallen heat lamp is a major problem.
Apart from predator issues, and maybe adjusting the heat lamp, that looks like a good setup.
We need to buy or build a coop for them in the same spot where the cage currently is. I have no idea about building a coop but am pretty handy with woodworking. I would just need some advice or suggestions.
For a coop in that spot:
Hardware cloth is a good starting point (the sturdy kind with holes 1/2" in size.)
I think you could make a wood frame, with a solid roof, and put hardware cloth on all sides, plus something solid for the bottom. The roof would provide shelter from sun and rain, the hardware cloth sides would provide good ventilation, and all of it would protect them from predators. The wood fence and the house will provide shelter from wind, so you don't really need to make solid sides for the coop.
Try to size it so you can walk in easily, or else so you can reach all areas inside. Chickens like to lay eggs in the corners you can't reach, and of course they run into the far corner when you try to catch them.
A general rule of thumb is that each adult chicken needs 4 square feet of floor space in the coop, not counting any space that has a feeder or waterer or nestbox. More space is fine, and the chickens are usually happier if you do give them more space.
You will want roosts for them to sleep on (probably between 2" and 4" wide), allowing about 1 foot per chicken on the roosts. They will all want to sleep on the highest roost, so you might make more than one roost that are all the same height (so they are all "highest.")
They will need nestboxes for laying eggs. Hens are usually happy to take turns in the nestboxes, up to about 4 hens per nestboxes.
(This is not everything you need to build a good coop, just the things I thought of when I looked at your photos.)
