Currently, LinkedIn isn't one of the options for sharing. However, as you mentioned, you can copy the link to share it there. Since there are many social media providers out there, I don't think every single one can be supported. The supported ones include Google+, Facebook and Twitter which are the major social media providers.
Yes, if you are sharing directly, you will earn the badge. However, you must use the link from the 'Share' button as it includes your userid.
httphttps://meta.stackexchange.com/q/289248/327561(327561 is the userid).
httphttps://meta.stackexchange.com/questions/289248(This won't work)