There are PWAs available already. You use https://nostr.kiwi and https://zapddit.com/. The former supports NIP-46, allowing you to log in with nsecBunker. Here’s my detailed guide on how to use this tool: https://lnshort.it/nsecbunker

The second question of yours is related to the Nostr protocol. It’s not yet sorted, but ima sure it will be.