I want to see the number of tag wiki that I edited till now. Also I want to see which are accepted and which are rejected with the detail of all those tags. How can I know that?
Also I want to know the progress for
Research Assisstant badge.
You can see the total number of approved tag wiki edits you have on the right hand side of any tag wiki page. It looks like this:
In order to see the other information you want, you'll have to troll through your "Activity" tab on your user profile under the "suggestions" filter. It looks like this:
There you can click on each suggestion, and look at whether or not it was accepted. If it was rejected, you will see who rejected it, and what reasons they listed for the rejection.
With a delay up to a month (or is it two weeks?), you can use the output of SEDE query Top Tag Wiki Editors (supposing you have edited more than 10 tag wikis) or write a more targeted SEDE query to see the number of tag wikis edited till now.
This also works after you get the current highest level badge (currently Research Assistant) - the progress indicator is absent (it will not show 100% or more and the current count) and on sites where the progress indicator is not enabled (like on this site, Meta Stack Overflow).