Question can also be a suggestion
Why doesn't MacOS have a search bar like Windows browsers,
Why can't my search be opened in the background or new tab？
Why can't translation software translate into a fixed language? Every time, it needs to be re selected, which is very troublesome
@Junhong Much of what you ask can be done through Vivaldi's Settings (⌘-,)
Check this box to search in new tab...
"Alternatively, type the search term in the Address or Search Field and press ALT + Enter to open the results in a new tab," says the Help page!
(Searching a page selection in a background tab is also a selectable option!)
I use the address field for searches, so I selected "Show as button" and then edited the button out (⌘-rightclick...)
And if you want a separate search box on the address bar
check this box to show the search box. with its own text field.. (You can also show the search engine's icon, if you want.)
Hope this helps some.
Of course, if you choose "Customize Toolbar" there are many options...
(Hover the tool-tip "?" in the upper-right for more info and ideas.)