Leaves are spotted yellow and green and just drop off. No curling of leaves. Do not detect any insect problem
There are several possible explanations for mandevilla leaves turning yellow. One of the most common reasons is improper watering. The soil must be kept damp but not soggy. Too dry and too wet soil will damage roots which then damages leaves. Another possibility is disease. There are several that attack mandevilla. Here is a good article describing these: https://www.gardeningknowhow.com/ornamental/vines/mandevilla/mandevilla-disease-problems.htm. It does not look like a pest problem from the pictures but carefully examine the leaves, especially the undersides, for tiny insects. You may not see them move so it has to be a very deliberate examination. It is always a "best practice" to remove leaves that have fallen. If they harbor disease, the disease passes into the soil or puts other plants at risk when the wind blows diseased leaves around.
I have a potted Mandevilla which I brought into the house for the winter. It seems to be doing well. At first it dropped a lot of leaves, but now there are new leaves and growth and it looks pretty healthy. The other day I noticed that one of the "branches" seems to have split down the middle and has white fuzz on it. Is this some sort of disease or nothing to worry about? I was really hoping to keep this plant alive so that I can place it out on the patio in the spring and have it flower again. It was an extremely healthy plant and bloomed way into the fall.
In most cases the plant can fight this off, but to be sure, I would recommend rubbing peroxide on the area to get rid of this. There won't be much further action necessary after this. I would also recommend letting the soil dry out slightly before each watering, so that the issue doesn't continue.
prefer mandevillas to diplidenaeas
In short, trailing plants don't like to climb and climbing plants don't like to trail. You can, certainly, grow them in container, but you will want to be looking to let them climb onto something. They will naturally attempt to hold themselves upward and fight the trailing nature.
This article will help: https://www.gardeningknowhow.com/ornamental/vines/mandevilla/mandevilla-plant.htm
how can I identify and treat the cause?
This sounds like it could be the start of a fungal or bacterial infection. For general care of this, as long as it is one of the many curable issues, and not an incurable one, is to apply wettable sulfur and dolomitic lime to the soil. You can also add a little peroxide with your water when you water.
Here is an article for proper care of these: https://www.gardeningknowhow.com/ornamental/vines/mandevilla/mandevilla-plant.htm
for mandevillas
Stake, trellis, fence or wall anchors? Check with a local garden center.
If you give more information on your site and growing conditions, size of plants etc we may be able to provide more specific suggestions.
The leaves also have green spots
This can have a few different causes. This article will help you to narrow these down: https://www.gardeningknowhow.com/ornamental/vines/mandevilla/yellow-leaves-on-mandevilla.htm
I want to put mandevilla outside in the garden, would I cut it back for the winter?
You are in zone 9a and this is within the growing zone for Mandevilla.
If your temps fall below a normal range for your area, you can add some protection with plant blankets; they are found at garden centers.
https://www.gardeningknowhow.com/ornamental/vines/mandevilla/mandevilla-ground-cover.htm