Advent are rarely if ever first in Trade/Resource Extraction unless they are totally dominating the map. It's not something they excel at. Planetary income is something they excel at though, thanks to the technology to boost max allegiance along with their awesome Temples of Communion. 
(Allure of the Unity, I think the tech is called. It's incredibly awesome.)
In terms of turning Tradeports into extraction focus ports, the rule of thumb is to only do it at planets with significant extraction. On an Asteroid with 2 extractors the extra resources simply aren't worth the value of a trade port. If you have a planet with 4 extractors and high allegiance, then it's a good idea to convert some over (you still want a Trade Port active for trade to keep the trade network going).
The Deliverance Engine is your super weapon. It doesn't do anything by itself. If you select it, you'll find a command to target it at a planet. If you target it at enemy planets, it spreads your culture there temporarily. This can give you line of sight on enemy planets (with a tech), allows your culture bonuses to apply when you invade, and if you bombard a planet enough can actually overthrow enemy colonies there. Just remember that you have to target and activate them yourself, or turn on auto cast after it's built.
