Full nodes are run by many users not just miners. In other words, every user who runs Bitcoin core and has opened outbound connections are running full nodes. See
https://getaddr.bitnodes.io.
So if a node checks the new block and rejects it, that would prevent propagation of it, right? So if (I'm just guessing here) a typical miner connects to 10 nodes when mining, and those nodes are responsible for broadcasting the new block out to everyone else, then only those 10 nodes would really need to ID the false transactions and fail to pass the invalid block along?
Thanks for explaining guys!
Also, regarding your other question:
is there a shortcut to verifying each send address is real and had the funds?
Yes.
Bitcoin does make use of a "shortcut".
Specifically there is no such thing as a "sending address".
Instead transactions spend and create unspent outputs, and every node maintains an indexed list of all the currently unspent transaction outputs (commonly called the UTXO list).
When you "receive bitcoins at an address", what actually happens is that a transaction creates new unspend outputs that are encumbered with a requirement to supply an ECDSA signature generated with a particular private key in order for those unspent outputs to be used to fund a future transaction.
When you "send bitcoins to an address" You supply a list of the unspent outputs that you are spending, and a valid signature for each of those outputs. Each node searches their indexed UTXO for each of the unspent inputs in your transaction. If you are using a UTXO that they don't have in their list, then they won't propagate your transaction. Then when each node receives a block, they check every transaction in the block in the same way as each transaction input is checked against the UTXO list, it is then removed from the list, and any new outputs created by the transaction are added to the UTXO.
Therefore, you can't create a transaction "1MickeyMouse34fg4... sending 10,000 BTC to 1YVEndj8D...". You would have to create a transaction:
List of inputs that nodes will find in their UTOX
Valid Signatures for each input
List of outputs created by the transaction
Scripts for each output that describe what the requirement is for it to be included as an input in the future