"I have previously imported the private key for a 1Archive... address into Bitcoin Core and noted then it added an address like 1AXbRq... and I didn't follow it up further."
First of all, I won't be answering the first question. It just sounds weird, and 99% it's caused by a mistake.
How does this work? I understand that there is an underlying 1... address behind a SegWit address but how would I spend any BTC if it is not showing in my wallet balance? What else should I do?
Here's how importing a P2PKH address works:
WIF result is affected by three parameters: Privkey type (a byte for BTC private keys), the private key (that also contains which
y coordinate the pubkey has, if we want the compressed pubkey), and an extra
0x01 byte if we want the compressed pubkey. There's only one public key that can be generated using that WIF.
Here's how importing a P2SH address works:
1) Dump the WIF of the P2SH address
2) Import the WIF
3) Make a transaction, and watch how your wallet or Bitcoin Core doesn't recognize the transaction.
It won't recognize the transaction, because there's infinite number of P2SH addresses one can create using the WIF (Note: script has a size limit, so it's not infinite)!
To inform your wallet about your P2SH address, you should also import the script. And, this is not supported by 99.9% of the wallets. (However, the wallets can import Witness addresses, as they're standardized and widely used) That's why Electrum doesn't see your P2WPKH address, if you don't tell it to use the Witness address.