If a website is cached, does it mean that it has been indexed too?
In this snapshot cache date is shown as 18th may but when i did site search for this domain I found 0 pages were indexed by google last week.
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When Google fetches a page, it is immediately stored within the index. This is where the cache comes from. The page is indexed.
When you do a site: search, this is NOT a reliable indication of whether the page is indexed or not. The reason for this is simple. The process is not simple. There is a fair amount of work that has to be done before a page will show up in the SERPs. Generally, this is not a slow process, however, it is also not immediate.
While the site: search directive is unique and often more complete, any SERP (search engine result page) list is a byproduct of quite a few algorithms regardless. All of the algorithms require metrics that may not have been collected yet. Some of these algorithms are filters that will prevent a page from being displayed. It is very possible that a page that has been fetched and cached will not show up in the SERPs even with a site: search directive. At least, not for a few days. Under normal conditions, this all works fairly quickly and works as expected.
I am making an assumption.
If you are using Google's Search Console to Fetch and Render / Submit to Index, please know this is an incomplete process. Because this feature has been abused, it has been modified. When you do a Fetch as Google / Submit to Index, it is temporary and incomplete. The regular googlebot will visit within 1-2 days and will make it all official.
You have to remember this.
Search engines are not real-time and do not work on your schedule. Period. You cannot hurry them up and you really should not try. With search engines, the best tool in the box is patience. We all go through the same thing. Your site is important to you and so is everyone elses. You are not the only worrying about how Google sees your site.
You are fine. Do not worry so much. Please. You are too young for that!