By default I use Google and it shows search suggestions, but when I add a custom search engine like https://www.google.com/search?q=, or a completely different one, I don’t get any. I don’t really care where the suggestions are coming from, they could be the same as the one for the default engine.
That’s a good question, after reading the documentation it seems the website that hosts the search has to support the OpenSearch suggestions, then, if it does, you’ll probably have to add it when inside the website with the plus icon that appears after clicking on the browser’s address bar, last time I checked I think there were a few differences between that and adding it manually by URL template.
I haven’t tested this however, so I can’t tell you which sites work with it