1. Question: Is it possible for the Bitcoin network to survive with only one node?
Bitcoin started with only one node, so clearly under the right circumstances bitcoin survived just fine with only one node.
However, given the nature of bitcoin's security model, it would be a centralized system where everybody would need to trust that one node for all information about the blockchain and would need to trust that one node not to act maliciously. As such, it really wouldn't be the same decentralized trustless bitcoin that we have today.
2. Question: What is the difference between a trust node and a distrust node?
I've never heard of a "trust node" or a "distrust node". In bitcoin as it is currently working, nodes do not trust each other. Every node validates every transaction and every block.
3. Question: What happens if there are more distrust nodes than trust nodes?
Are you just making up combinations of words and hoping that they have meaning? What is a "trust node"? What is a "distrust node"? In bitcoin, nodes are never supposed to be trusted.