I tried to root some inside in water, and they didn't wilt, but they didn't root, either! ):
We would like to grow some on our see-thru fence, as it is evergreen, and the road is getting much busier, with more people moving into our neighborhood, and more kids walking to & from school . . . Plus, it will provide some welcome summer shade for our resident (& visiting) wildlife, a cooler micro-climate for some plants, as well as our dog.
I just rooted some cuttings of Bignonia capreolata, so can offer this advise, based on my experience:
How thick must the branches be to root them successfully?
Most any diameter of cutting will root, except for the wispy and thin end of the vine.
How warm must they be kept?
Your ambient temperature will be fine - providing humidity is the most important thing in rooting.
Is it best to stand them upright, or lay them flat like a ground-cover?
Just loosen the soil right where you want them to grow, stick them in the soil and they will root. You will need to provide good shade because the direct sun will dehydrate them quickly and they will die.
Should I remove 1/2 or more of the leaves, or, cut them in half, or both?
You only need to remove the leaves up to the 4th set of nodes (where the leaves attach to the vine) and keep them moist. They should grow roots in about 3 weeks.
I had very thin cuttings to work with, and they had been in water for 3 weeks without rooting (or wilting), so, I put them in some potting soil mixed with earth from where I intend to plant them eventually, & put them in pots. I sprayed them several times daily with water, and kept them out of direct sunlight, with some plastic surrounding them to help hold in moisture. I now have 4 stems which allow me to gently tug on them without coming out of the ground. (: (The thinnest ones died, as expected.)
Since we have more gopher activity, I am not eager to plant them in the ground. Maybe transplanting them to larger pots --for now-- will be safer for them!
We've been having much colder weather at night, so that 2 of the 4 have lost their leaves, but I'm not sure if their dead, or just dormant. I've moved them to a more protected area for the winter. If necessary, my neighbor says he has plenty of stems I can use for more cuttings next spring, which is when he recommends trying to start them. That way they can grow throughout the summer.
He has this plant on his chain-link fence, and it grew along it. over his drive-through gate, which he still drives his care in & out of on a regular basis! I never saw a vine that was covering a moving gate, before!
Our winter low was an unexpected 16 degrees F, and I lost my rooted cuttings for sure!
Got some new cuttings rooting now, though. This time I cut them in sections with 3 nodes each, and left two leaves at each node on the same side. I laid the other side in cacti/succulent potting soil mixed with some sand, in a see-thru, hinge-covered, sandwich tray. The soil was moist from the start, and before I clicked them closed, I put a stick in a short section of plastic drinking straw in the center under each lid, to allow some air in & out (though not much). I've kept them inside for a few weeks, at between 70 & 80 degrees. Two of the 5 plantings are putting out 1 new stem each, and I have them shaded on the hot front porch . . . they seem to be doing fine. (No wilting at all.) The other 3 are still in the house. My sis in-law just today sent me several more cuttings, with thicker stems. (I just used near the tips for the ones already rooting!) I pruned & misted them, & they await my attention in a plastic grocery bag!)