I’m on version 2026.2.0 and the search state doesn’t persist across app lock. I agree with @grb : AFAIK the desktop app has never persisted search state or folder state across app locks — see the discussion here.
The closest thing added over the last year is the browser extension persisting some states across the extension pull‑down/pull‑up, but that’s also time‑limited.
So on point b) I wouldn’t hold out much hope either, since there doesn’t appear to be even a feature request.