Unfortunately, there aren't any set guidelines or specifications to building cham cages and most of us just read up as much as we can on this forum and then stumble our way through it lol. Try doing some searches for "cage build" or "DIY cages" and I'm sure you'll find out most of what you're looking for, that's what I did. Thankfully there are a lot of members on here that have shared their trial and error experiences and there's always a handful of us on here that are willing to help out so try not to get discouraged.
Now to answer your question about the screen... The best way would probably be to sandwich the screen material between your frame and a thin trim board or staple the screen down and use the trim board to cover that up. You can also be fancy and pick up some decorative wood molding (like for a staircase or built-in book case) and use that to hold the screen and give it a fancy touch. I wouldn't worry about any type of sealant though since the polyurethane should soak in to any gaps and help fill them in. Just be sure to get a furniture-type polyurethane that can be used on interior woods like baby cribs so you know it won't contain any harmful chemicals. Or, you can do what I did before I did some more reading and thought some of that up and just use a combination of the metal corner braces and wood staples to hold the screen in place lol. Once I had it tight and held down I used some exterior door/window caulking to cover up any fraying on the screen edge and make sure there weren't any gaps that any feeder insects could escape through. It sounds a little hokey but thanks to some masking tape it turned out really clean. I plan on starting the adult size cage for my panther, Loki, in a couple of weeks and will be trying a new hybrid screen design. Basically, it will have a 1x2 oak frame and I'll be router-ing a channel on the inside edge to slightly recess a black aluminum screen rail and then install the screen in to that like you would a normal window screen. It's more work and I'm not sure how it will actually turn out but I think it will give the cage a unique (and cool) appearance and I will also be able to easily replace the screen should I need to, unlike the other methods I just mentioned where the screen is more or less permanently installed.
As far as wood goes, I used red oak 1x2's from Lowes to make my cage since I wanted it to look like a piece of furniture that matched what we had in our house already. It turned out really nice especially once I put some dark cherry wood stain on it and got a couple of coats of polyurethane put on it. The oak was also pricier than some of the other building material options. Most of the cabinet-type building materials like (pine or oak 1x2's) should be safe and are generally finished a lot better and are usually baked to get rid of/dry up any sap unlike regular construction-type lumber that can leach it out over time. Like Wykd said though, be careful with treated lumber as it contains chemicals like arsenic that can definitely be harmful. Also, be sure to let the cage air out for at least a couple of weeks in a well verntilated area (preferably outside) so that the fumes from the polyurethane, paints, sealants, etc. will have subsided before you put your cham in it.
I also made my own faux rock wall with planters but I didn't build any branches in to it like others have done, mainly because when I built it I wasn't sure what types and sizes of plants I was going to be putting in there and I didn't want to have to work around what is already in the cage. A simple way to attach branches later on is to press the ends of the branches against the screen and then stick thumb tacks/push pins in to the ends to hold them in place. Another option since you're using wood is to use appropriately sized screw-in eye hooks (they make rubber coated ones so you won't have to worry about the zinc accidentally being ingested) that you can screw in to the wood and then stick the base of the branches or sticks in it that should hold them securely in place.
I'm by no means an expert but hopefully some of my ideas will help you out. Here's a link to my build thread in case you want to see more of how I did something:
https://www.chameleonforums.com/my-first-cage-build-84441/. Good luck and be sure to let us know how it turns out and if it turns out great you can make your own build thread so others can learn from what you did.
