Trade ports give you an increase in credit income.  Every trade port in your empire (unless the target planet has a bonus, like +10% trade income) gives the same amount of benefit.  However, if a trade ship is destroyed, its home port suffers reduced income for a short duration.  The amount of money each trade port gives you is based on the "longest chain" of trade ports in your empire. 

Refineries simply increase the amount of resources produced by all resource asteroids in their area of effect.  For Advent, this is all resource asteroids in their own gravity well.  For TEC and Vasari, this is every asteroid in the gravity well you placed the refinery, and every asteroid in adjacent gravity wells as well.  Each asteroid can be affected by only three refineries; any more and they stop giving benefit.  However, the Advent resource focus doesn't have this limit, and you can keep adding more beyond that point.  However, it is still the worst refinery by far.

Most of the time, trade ports are always better than refineries.  However, sometimes placing a trade port will interfere with your longest trade route (if it provides a shortcut across your empire, for instance).  Other times it's literally teeming with resource asteroids.  In these cases, the refinery is a good investment.  Typically players invest in lots of trade ports early on to build a high credit income, then add refineries later on.

If you mouse over a the box that  display how much money you have, you get a breakdown of all the income you have. Also a white line will appear on the map that is your longest traderoute, making that as long as possible will give you more money

If you mouse over the metal or crystal parts of your display you get a breakdown of all the crystal or metal your getting

if you mouse over a particular extractor, crystal or metal, you get a box that will say how many, 0-3, of 3 refiniries are being used/helpful to that extractor.

I haven't moused over a trade port itself, that would be interesting wouldn't it?

You can add trade ports to place that do not extend your longest trade route. They will add income. and Yes trade ports can "stack" putting 3 there does give you 3x the income of 1.

Regarding the planetary pop upgrades:

Non/barely-upgraded planets cost you credits over time.  Upgrading will eventually reduce this to 0.

Upgrading will also increase the planet's maximum population, which will result in higher tax revenue as the population grows.  If it's far away from your capital, the allegiance will be lower and you won't get as much revenue.

(* Note:  Advent in base game can do research that increases the maximum allegiance of their planets within their culture; in Entrenchment, they can also build a starbase and use two modules to increase maximum allegiance by +15% each.  That's 15% ea. absolute, not relative to the normal maximum allegiance.

Advent capital + two of the allegiance modules + max culture bonus of 10% (two levels of research) = 140% allegiance = 140% tax revenue from that world.  Doesn't affect trade ports, apparently.

)

Try immediately researching assailants / javelis / illuminators (depending on your faction) and making these units instead of your basic light frigates.  The AI doesn't know how to stop these units (even though they can easily be researched just with your starting resources) early on, and you can cream it in early battles, gaining the upper hand if not winning outright then and there.

The hard AI gets a resource cheat, but is rather careless.  If you can use this to your advantage, it will just get its units killed over and over again.  Then instead of building up a huge fleet of doom, the only thing building up is your capital ship levels.

Desert planets have by far the most logistical slots of any type of planet.  However, there isn't any particular reason to use the extra slots in particular way just because it's a desert planet.  You could put labs there and build trade ports elsewhere to the same effect, it all depends on the bonuses on those planets and the phase lane layout for longest trade route.

Don't forget two levels of Allure of the Unity for a max of 150% at home planet.  It takes a lot of investment in research and starbases but very late game Advent economy now kicks some serious butt.