Lite allows you to have a 7 day trial for $7. After the trial ends, it costs $99 per month or $990 per year. This plan allows you to track up to 5 projects (websites) and track 500 keywords per project. You can search 25 batches of keywords per day, but you can search up to 10,000 keywords per batch. I like to use it to search multiple ideas for long tail keywords at once so I don't use up all my searches.
Basically, Google shows the autocomplete suggestions whenever you start typing anything into Google search box. It is in Google's best interest to show the most relevant keywords in the autocomplete suggestions. Keywords that would help Google to retrieve the most relevant websites and help users find the most relevant content for their search query.

Ahrefs make use of two link updates in your website - the Fresh Index and the Live Index. Fresh Index which updates at a longer time frame. It shows the dead links on your website, giving you the chance to change or reclaim the dead link. The Live Index, on the other hand, makes link updates every 15-30 minutes. This provides you with the chance to see regular link updates on your website.