I've created a new Segwit wallet which has the addresses starting bc1. Am I correct in thinking that any Electrum wallet that is version 3 can send a transaction to these addresses but no other wallet can?
I imagine that every address/wallet can send to this kind of address. But most of the services and wallet won't support it by now, which makes it impossible to withdraw from your exchange account, for example.
Secondly, (this is the bit I'm more unsure about) I can send from that wallet to all other addresses as the transaction is processed on the blockchain not by the receiving wallet?
Yes. You can actually see this kind of address in the transaction if you check the tx ID at Blockchain.info; You just need to enable the Advanced mode.
Here is an example:
https://blockchain.info/tx/7b269691a2716da306d88cfc08e088e176654148c2b79f3c94c62ffca657c038