Is there a tree whose branches intertwine together? in the sense of forming braids and also if there are trees whose branches literally going through each other.
Not in any ordered sense - trees may produce crossing branches, but not to any particular shape. You're probably thinking of trees which have been shaped, pruned and trained, either by braiding, pleaching or other formations, to make certain forms. Information on pleaching is contained in this link https://www.gardenista.com/posts/gardening-101-pleaching/ but for more general tree shaping, this Wikipedia article names quite a few methods https://en.m.wikipedia.org/wiki/Tree_shaping
Ficus macrophylla, the strangler fig, grows aerial roots which naturally intergraft See here https://en.m.wikipedia.org/wiki/Strangler_fig