If I were to graft two apple saplings together, by bending the tops toward each other and lashing them together, will the plants grow as one and benefit from one another, or will they be fighting each other for root space and light? If they would grow with each other, then I could theoretically grow a line of closely spaced fruit trees to any length, and they would be strengthened by each other in bad conditions.
There are a couple of answers to this question. Especially where trees are concerned, you can graft two or more trees onto the same rootstock, or even a single limb into a tree.
But if the graft takes, it won't behave too much more differently than just more branches of the same tree. Structurally intertwining them will not be different than if you had just taken a single tree's branches to support each other. The graft will usually only have a single set of roots, from the host tree. They will not compete. The tendency will be for the branches to grow apart so that they can independently get their own light. This is very much like any other single tree. Not sure about fusing two halves of a tree together - exposing the roots would tend to kill the tree or unsettle it.
If two trees grow close enough together so that their trunks touch each other anywhere along the length of the tree, then they will eventually fuse. This generally only happens at the trunk because, unlike small branches, the trunk really can't be pushed out of the way as easily. It doesn't necessarily need to be two trees of the same species either.
There used to be a fused sycamore-maple on my school campus (it was damaged in Sandy and was cut down). They weren't completely fused together, but you could see a joint at the base and about 20 feet below the canopy where the trunks essentially became the same. There was no distinction between the two separate trunks.
But to finally answer your question; when the trees fuse they pretty much become conjoined twins. I'm not sure if they transfer genetics to each other, but they do share resources.