Can I connect to a node as read-only? In my Electrum wallet I can open the Network Window and view Nodes I'm connected to. Here's an example: electrum4101870.fullhealth.net. I can also find public nodes at https://bitnodes.earn.com/nodes/leaderboard.

Is my wallet streaming data from these peers? If so, how?

My understanding is that in order to stay connected to a node you need to exchange messages about the chain...

Or, is there a way I can connect to a node and just read outgoing information, such as new blocks + transactions.

Thanks
It is important to note that your Electrum wallets connect to Electrum servers which speak a different protocol than the Bitcoin P2P network. Electrum servers are full nodes with additional software running which provide the data and protocol that Electrum uses.

If you want to be strictly read only, as in you never send the node any data, then no, it is not possible with either the Electrum protocol or Bitcoin P2P protocol. Both protocols have a handshake at the beginning of the protocol where both nodes (or client and server for Electrum) exchange messages before the rest of the connection can continue.

Furthermore, both protocols have ping messages which have to be responded to. If a node (or client) doesn't respond to the ping, the other side will disconnect you. So if you have no send capabilities, you will not be able to maintain the connection for very long.

However if you just mean you aren't sending any blocks or transactions, then yes. You can most definitely connect to a node and never send it any blocks or transactions.