There aren’t a lot of tools like that as far as I know, although some people have their own spreadsheets, IFTTT recipes and so on and they use them for analysis. Perhaps some will see your topic and give their tips.
Within the panels we have, it’s mainly to go here: https://www.fiverr.com/users/wordandrecord/manage_gigs
(Other people reading this will have to put their own username where “wordandrecord” is)
and read this: http://support.fiverr.com/hc/en-us/articles/201733423-What-are-Gig-stats-
If you have a gig that isn’t functioning well and you don’t mind messing with it, you can also trial and error some things. When you do this, the editors may look at the gig since it’s been changed, so it might have a day or two of down time.
To do it, you just search on the main page for gigs like yours by keywords or filters that you think people would use to find what you do. Take note of the title, tags and keywords the first results have in them. Try applying similar ones to your own gig, then check it in search and in your gig management page to see if your changes push your gig up or down or change the stats in a favorable way.
As far as external tools that work with Fiverr, one example is Adam who has published some IFTTT recipes that might be useful: