I think something similar has already been requested numerous times in various meta posts, as well as chat. Can you elaborate on what the problem is? It's very much clearly available on desktop, but I do agree hiding it behind the hamburger menu, as is done in the mobile view, is the wrong choice.

For your second suggestion; I've already brought that up, at least in chat, and added it to my notes, but I'm unsure if it's been mentioned on Meta. If anything, it belongs in a separate post from this, or a larger search overhaul Q/A, which I think there might be. The search results aren't always the best, so I can understand if you struggle to find existing posts.