So when is the best time to plant a fig tree in ground?

Since I rent, this decision is hard for me, but if I can take a cutting when I leave in a year or two and my collection looks like it is going to grow since joining this forum, I may be willing to gift my first tree here for future tenants to love.

I was told it was 3 years old when I bought it and so that would make it a 8 year old black mission and it produced 30 figs this year which to me is amazing. Its starting to take a branching shape but all the branches are pencil size. I probably have been doing something wrong, hence why I am here and am wondering if I plant it in ground if both me and my tree will be happier?

If I plant it, will it just take off for me to enjoy this coming year? Do I need to root prune it? Any special things to take into consideration when I do plant it? Or is it better to start with a new tree instead of this one since its been in a container its whole life and may be stunted?

If I were you, I would plant a newly rooted cutting, or root one in place, before it gets hot. The older tree might take off right away but will end up smothering the existing branches with vigorous suckers, it would probaby increase the crop but they might not have time to ripen, or the vigorous growth could prevent figs from forming at all. There's no way to know what will happen either way. 

Older tree: 
Better able to survive a bad planting situation.
Needs more water and time to become established, bigger planting hole, more likely to be stunted by girdling roots. 

Younger tree
Less work all together, less investment, more likely to grow into a big healthy tree faster.
More vulnerable to poor soil conditions.

And think of it this way, if you learn only how to plant 8 year old trees it will be a very long time before you benefit from that experience again. Get good at planting young trees and you will grow as a grower much faster.

It sounds like you have an 8 year old tree and only produced 30 figs in Los Angeles? Put it in ground and let it do its thing you will be way better off