-
Not quite.
When you use WhatsApp Web it treats the WhatsApp Web client as a new contact of yours, and so it (the browser) creates a new public/private key pair for that contact and then requests all messages be resent from your mobile to this new contact E2E encrypted so it has access to the messages.
Theoretically only E2E encrypted message data should be sent between client and browser (plus over TLS for extra safety), with the private key having never left the browser, but clever people have analysed it and worked out that things are somehow retrievable from just the HTTPS data sent back and forth.
Depends who "them" is?
It's generally thought of to be secure (but hard to verify this).
One thing for sure, if you use the Web interface then it's pretty safe to assume that the UK Government has access to the private key of their HTTPS Cert, and so all traffic to/from that site would be readable.
I'd trust [EDIT]
TelegramSignal a lot more than Whatsapp.[EDIT] Just for starters, Whatsapp is end-to-end encrypted, but: