Is there a way to try out my Pokemon moves if I am home, not next to a gym?

No, there is no way to battle if you are not close enough to a gym. Trying out your moves is also not possible any other way.

Of course, Niantic might add a new feature in the future, but currently there is none.

While you are unable to simulate a battle to see how moves look, you can see a move's stats and attributes to determine what best suits you or your needs.

To see a list of stats for each move, you can look here. However, since the moves' damage and subsequent energy increase has been changed, this site has outdated numbers. To see updated energy gains, look at this post.

No, but you can compare the moves you got with other moves using this table:

https://goo.gl/SEBo2y

Here you can check which pokémons movesets are the best compared to other movesets that all other pokémons has.

In the episode below, he explains how to use the spreadsheet and why is that so important to use it.