Hello,

I've had Veiled, carpet and Jackson chameleons so here is my take on your questions.

You will find many "styles" of caging that keepers use and all (ok most) work well. Most will have their pros & cons.
An adult Veiled (male) should be in no less than a 3'h x 2'w cage. Larger is better but it can be a challenge for space for the keeper. The most important part is the height of the cage (chams are arboreal) and good ventilation. I really love the cham cages from LLL. They are sturdy and very easy to clean, but yes they can be expensive. If you are thinking of building your own you have the luxury of designing it to fit all of your needs. A google search can find you many "how to" or designs of home built cages.
The size of the screen you use is really based on the feeders you use and where you keep the cage. You really do not want a bunch of crickets running around after escaping from the cage With that being said I have seen cages with vinyl window screen (works very well) to the plastic mesh you use in a garden etc. Again all can work based on your needs.
The only problem with using wood is you will need to seal it before using. Chams in general need high humidity and that is mostly achieved by frequent misting. Untreated wood will mold up and rot fast. Don't be discouraged by this though because you can make a very nice functional and good looking cage with wood. Some people use PVC as the frame and attach some sort of netting with zip ties. You do have a few options with what you use. My advice would be to look at as many DIY cages as you can and find out the pros & cons. Then decide what works for you and go from there.
Veileds are one of the few chameleons that will actually eat the plants so you need to be careful on what you use. Your 2 choices are great for a Veiled. I would suggest going to any nursery/Lowes/Home Depot and find the healthiest plants you can. Then take home and get rid of all the soil in the pot. Wash off the roots and replant in a pot that works for you. Careful on what potting soil you use, not only in reference to fertilizer but also how clean it is. Nothing is more of a pain than having an infestation of gnats or other tiny bugs coming out of the soil. Mild dish soap can get rid of this and you can use when you first wash the roots.
I used to have a lt of both plants and would rotate them in and out of the cages. When not in the cage I would put them near windows or outside to keep them healthy. (The lighting you use for your cage will also help the plants.)

I am excited for you! A male & female veiled were my first chams and I got them when they were around 5 months old. Males are a good choice because they are more colorful and you do not have to worry about them becoming gravid.
Anyway, I know there are others on here who have experience so you are in good hands. Post some pics when you get yours!

Jim