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.
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.