Can a Rinnegan user use Mangekyou abilities? Since Rinnegan is an "upgrade" of Sharingan, then it should be possible that all the techniques of Sharingan will be carried away to Rinnegan like the Mangekyou.

If not, then why?
To put it bluntly, no, the rinnegan cannot use mangekyou abilities. As you can watch in the anime, Madara can switch between Rinnegan and Sharingan because he awakened both.
Nagato only had the Rinnegan.

Basically, the Rinnegan isn't an upgraded Sharingan, it's what comes next, but it's a completely different eye.