Could a contract be programmed such that the redeemer seemingly comes from nowhere from what the node / someone monitoring the blockchain could tell? I am thinking if you gave the contract some ada and had it pay fees automatically out of itself, you could run totally anonymous logic. Are there restrictions preventing you from doing this?
You seem to be misunderstanding how smart contracts work on Cardano.
Does the redeemer have to be from a wallet address(?)
Redeemers don't come from wallet address, they are included on transactions. And transactions must be created by a user off-chain.
I am thinking if you gave the contract some ada and had it pay fees automatically out of itself...
There is no such thing as automated smart contracts in Cardano. The datums at script address only change state as a result of a transaction submitted from off-chain.