@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.