Quote from: TheQuin on November 08, 2017, 08:43:40 AM

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?

Yes, Electrum 3 can send to this address from any wallet.

The support for bech32 is indeed very weak but Bitcoin Core can also send to (and receive) this address if you compile from github source. The bech32 support was included in commit aa624b6. There might be some other wallets that support it because these transactions can be spotted in the wild from time to time.

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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?

Support for these transactions is defined in BIP141 (Segwit softfork), so all the miners and all the nodes that understand segwit will process it correctly. Full Segwit support was added in Bitcoin Core 0.13.1 and these nodes and above will validate and forward these transactions. Being soft fork it is still valid in earlier versions but earlier versions will think that anyone could have spent such transaction because they ignore the witness signature.

I haven't tested any other software except for Electrum and Bitcoin Core. I sent and received P2SH-P2WPKH transactions on mainnet but I tested P2WPKH bech32 only on the testnet. Works both ways between Electrum wallets, and from/to Bitcoin Core with aa624b6 commit. Check the testnet to make yourself comfortable with it.