Let me start by saying that if you’re using a different browser, and expecting major UI elements to be the same as your old browser (without having to use extensions, etc.), that might be a bridge too far in terms of expectations.
Also these issues don’t seem to be Linux-specific, but I’ll jump in anyway and help if I can.
First, any interest in using extensions or no? You might be able to get the ‘search bar’ implemented through one, but I don’t think you can without that.
Likewise with the Menu bar. When I was a Firefox user I always turned it off anyway so it didn’t bother me that Brave (and Chrome, etc.) didn’t have one – and I can still use the keyboard to access the menu (Alt+E) if needed.
But, if you want to type your search into the URL bar and have it open the search in a new tab, press Alt+Enter after typing your search. Maybe see how that works for you.
For your ‘switch to new tab,’ try Ctrl+Shift+Click on a link.
It was quite an adjustment for me too when I switched, but I felt like the privacy advantages were worth it. I initially told myself I’d try it for 3 months solid, and once I got settled in, the differences melted away and I’m happier running this browser; no thoughts of going back.