Hi,
I did several major site migrations in recent months and if you get the basics right, there should be no (negative) impact on your rankings. It's a myth that site migrations always lead to a drop in rankings. Changing technology should not have an impact - Google is all about the user experience, not about the technology used to create the user experience. Having a responsive site should increase your traffic (and rankings).
Not affiliated to them - but as already indicated by Travis a tool like Screaming Frog (the paid version) is almost a pre-requisite for a successful migration.
The main thing you should take care of is that all your current url's are properly redirected to avoid 404's after migration. You indicate that the url's are not going to change so this should be no problem. To be sure - crawl your current website with a tool (like ScreamingFrog) to get the full list of url's. Export the list of url's & replace the current domain with your test domain and crawl the responsive version based on this list. Check if all url's return a 200 status (if they return a 301 - check if the redirected url's return a 200 status). As a side effect, crawling your current site could also show you other areas of improvement like images that are too heavy, endless loops, unbalanced site structure, etc. If you encounter issues with your current site - try to solve them with the migration.
Compare the main on-page items between the two sites (H1, page title, metadescription) - in your case they should be identical (if you use ScreamingFrog this is a pretty basic task - just export the craw results to xls & see if they match)
Crawl your new site - check if the number of pages crawled is identical (or close) to the number of pages crawled on the current site. Screamingfrog allows you to see the site structure - compare it with your current site, to be sure that important content is not pushed deeper into your site.
Use a tool like webpagetest.org to check the speed of your site - both for your current site and your new site to be sure that with the new designs you are not slowing down your site - do this for a number of typical pages on your site (hint: look at your most popular landing pages in Analytics). Do the same checks with https://developers.google.com/speed/pagespeed/insights/ - see if you improve the results with your new responsive site. Your score for mobile should be a lot better - if not - you did something wrong with your re-design.
For your most important pages (probably your most important landing pages / most visited pages) - do some visual checks between old & new site - is all the content present, does it look ok, ...etc.
I don't know if you get a lot of traffic from image search (check your webmastertools). If this is the case: if the url's and/or the names of the images change - make sure that the 'old' images remain accessible under the old url (in my experience it takes a bit longer for google to update the index for Image search).
Make sure that your sitemap is updated when migrating. Probably you block your test site for indexing - don't forget to remove this when putting your new site online.
After migration, check your webmastertools for 404's - idem for Google Analytics. Correct 404's as fast as possible (make sure that your 404 page is tagged in Analytics).
To answer your point 3 - if something goes wrong during migration, in my experience you should count on approx. 3 months for recovery, but with no guarantee that you'll return to previous positions, so better make sure that you get the points above right from the start.
Good luck with the migration!