If I graft two trees together while young, will they grow as one plant?
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.