1. I’m not really understanding this – is that a big gap under the slideout menu? Does it fill in ever? What browser are you using?
2. This is an iOS and sticky element issue when it comes to keyboards. I’m actively working on it, and have a solution implemented if your sticky effect is set to “none”. Don’t have anything for the other effects yet.
More on this topic here: https://generatepress.com/forums/topic/search-field-sticky-menu-problem-on-ios/#post-152405
3. This is a future planned option 🙂