Also having this same issue. Only in Vivaldi. I tried to remove keyboard shortcuts, just in case, for the actions. Still goes back as if it's "rewinding" or "fast forwarding" instead of going back or forward just one page.
@djanci: You can get to that same setting in Vivaldi, by going to chrome://settings/search#pass
If you were to disable and re-enable it there, I guess it's possible that Vivaldi would act differently. Of course it should not be having the problem in the first place, and none of my 14 installs do have the problem.
@dLeon has already answered your first question. You can turn that on in the settings here: vivaldi://settings/search/ (open this link in a new tab or copy & paste it into the address bar). Though I can search directly from the address bar, even when "Search in Address Field" is unchecked.
You can use other search engines in the address bar, by using their nick names (you can set them yourself). You can find out more about the search functionality in the Official Vivaldi Knowledgebase. You can also check out this tutorial.