It can be a long process. Start by editing a few Wikipedia pages with relevant content on topics you know something about, but without adding links. Do this a couple of times every month to build up some history on your Wikipedia profile. In 5-6 months, you can then edit or insert a link on one of the pages you previously updated. Just make sure the link points to a page with good and updated content, as someone mentioned earlier.

If your backlinks don't get indexed without help, they are not good backlinks. So let Google and other search engines find them themselves, and use your time to create good content and build more links.

Patience is key. It often takes time for search engines to discover and index new backlinks.

That depends on how often Google usually crawls your website and what header code you have sent during the downtime (404, 505 etc). But the quickest way is to ask for a crawl in Google Search Console.

In Search Console under -> Settings -> Crawl stats. You can see how many times and how often your site has been crawled.

Are you making the page for visitors og searchengines? Make it intuitive and with all the information needed to describe your tool. A better user experience the more links you will get. Just make sure you have internal links, so the linkjuice will flow.