The free variant for free backlinks is Google Webmaster Tools (now Search Console). It supposedly gives you a sample of all your backlinks, as well as a register of your newest links, which you should check regularly.
I personally suggest Majestic, as it's an invaluable tool for any serious SEO.
As for free ranking checker, I can check the ranking on any website for any keyword in less than 5 seconds using only Google Chrome.
Here is how I do it:
Go to Chrome and open the settings
To to "Manage Search Engines"
Scroll to the bottom and you'll find three tabs to fill.
The first two fill however you find it reasonable, the last you need to paste the following:
https://www.google.co.uk/search?q=%s&pws=0&hl=en&gl=uk&num=100
This formula will remove all personalization from your search (e.g. history, google plus, etc.), then apply a filter for language and location, in this case english and UK and lastly make the search results page list up to 100 results. Then you can make this the default search engine for your browser.
Then, go get this extension for Chrome:
https://chrome.google.com/webstore/...ension/nedmifdnblopopbcmdcgflboaclchebg?hl=en
It shows you the current number for each result, as well as the change in ranking from your last visit of the particular search query.
Type a keyword in the address bar, hit enter, hit ctrl + F and search for your website. If you don't find it in the top 100, you can safely assume you don't rank (different than not being indexed).
Setup takes 5 minutes and then each check takes only seconds. You get the most accurate results, you skim the SERP for all your competition, and it's free.