To migrate SharePoint from 2010 to 2016, since there is no direct approach for migration apart from moving it to 2013 first and then migrating it to 2016. Will Microsoft provide a temporary 2013 license for migration or do we need to buy 2013 license as well?
I disagree with Anand's recommendation. From your question, it sounds like you're ok with and planning too use the database attach upgrade method. If that is the case, I recommends that you simply download the fully functional 180 day trial from Microsoft here: https://www.microsoft.com/en-us/evalcenter/evaluate-sharepoint-server-2013
It's a VM img file format. This will give you 4 months to complete the double hop upgrade.
The best way of migration is to use any Migration Tool. I would recommend, Saketa Migrator as it consists of many features and as per the client requirement, they will customise the price and the product. All you have to do is register for a trial version which would be of 7 days(which I guess is quite enough if you know how to manage :P) and download the app. Just paste you source and destination and click the play button. Its quite easy.
You can do the same with the other migration tools too and would be cheaper than buying the SP2013 license.