Specifically, consider a network with 1000 mining full nodes and 1000 wallets. Now add a non-mining full node. Is the network somehow more secure?
Yes. Assuming that you are talking about 1000 SPV wallets, full nodes will be important. SPV wallets rely on full nodes to deliver accurate information and will have to trust them. This exposes them to sybil attack where a lot of nodes are created by a single entity and are modified to follow a different chain or different rules. The more nodes there is, the more expensive it will be.
SPV nodes verify the PoW. They don't have to trust their peers apart from having at least one peer that doesn't withhold the blocks or the branches. The only thing these full nodes can do is
not pass a block they consider invalid, but the SPV node would then still receive it from a miner.
The full node is a checkpoint on your own driveway. It doesn't secure the rest of the network in any way.
It does. They are in charge of verifying and enforcing the consensus rules. They help both the network and the light wallets in a way.
All a full node is "in charge" of is which blocks it accepts. All it can do is reject a block and not relay it. But the network is extremely well connected. Blocks will find their way anyway.