After a hard fork, what happens to the full nodes? Say 11,000 BTC full nodes existed prior to the fork. Do all 11,000 end up supporting both BCH and BTC? Or does BCH start building it's own network of nodes. Not asking about miners, just about full nodes (although, it will help if you can address the mining nodes as well).
A hard fork means that a new set of consensus rules is being used which splits the blockchain into a blockchain following the old rules and a blockchain following the new rules. There is no "in between" for a node to follow both rules and nodes are not typically equipped to simultaneously support two blockchains.
In the event of a hard fork, the nodes which support the new rules and the hard fork's activation rules will separate from the nodes which do not support the new rules. They become their own network and have their own blockchain. This applies to all full nodes, which miners are a subset of.
So in your example, none of the nodes will support both BCH and BTC unless they were specifically designed to do so. The standard behavior (i.e. nodes using Bitcoin Core for BTC or Bitcoin ABC for BCH) is to only support one chain whose consensus rules are hard coded. Some of those 11,000 nodes will have the BCH rules so they will split from the other nodes which do not have the BCH nodes.
Also, in the softfork scenario, I believe the 11,000 nodes are divided - say 7000 support the softfork (e.g. Segwit) and the remaining 4000 do not. In that case, how do transactions remain legit between the supporters and non supporters.
A soft fork is backwards compatible with old nodes. It must be designed in a way such that old nodes will not be kicked off of the network after the activation of the soft fork. So for soft forks, the soft fork was designed so that transactions under the new rules are also valid under the old rules. For segwit, this is done by stripping out the segwit specific components which are completely new so that non-segwit nodes are able to parse and validate the transactions.
In a soft fork, there is no chain split and all 11,000 nodes are still using BTC and following the same blockchain.