Military costs money and its effectiviness depends on the population size. Several planets with managable hazard levels will grow fast enough and will not have very large upkeep for the High Commands. Resulting in the military powerhouse.
This.
The self-sufficiency thig is entirely optional, and dependant on your preferences.
I favour it because being able to meet your demand for all resources with your own mining outputs reduces the maintenance you pay on all your buildings.
You don't even have to go all-in to get this either. Any resources you can supply yourself with will give you a proportional cost reduction.
If you can supply all the resources yourself you'll get a 50% discount on colony upkeep. (Hint: You'll need a free port to achieve this.)
Having everything all in one system also greatly reduces the occurance of shortages due to lost shipments, and is significanly easier to defend as you can overlap multiple military bases even on hostile worlds as they cost so little to maintain.
The minimum levels for resources depend on what kind of skills you have access to.
If you do not have the Industrial Planning skill which provides extra outputs then you're looking for:
Food: -1
Ore: +2
Rare Ore: +2
Volatiles: +2
Organics: 0
If you do have the the skill, OR you have hired an admin who has the skill, OR you have used an alpha core as an admin, you're looking for:
Food: -1
Ore: +1
Rare Ore: +1
Volatiles: +1
Organics: -1
Bear in mind that admins and cores can only look after 1 planet each, while you can look after several.
I think using an alpha in an industry also gives more output so you could potentially go even lower, but I don't use cores so someone else will have to confirm.
The Synchrotron can be installed in a fuel production industry, which it boosts by +3 fuel produced.
It does not affect the input requirements in any way.
Fuel production on its own makes a decent chunk of money. Adding a Synchrotron makes you noticably more.
Duzahk has only one gas giant and one small barren I haven't bothered to survey. It didn't seem in any way viable to me.
Those are the only guaranteed planets in Duzahk, and if there's nothing else there it's usually not worth bothering with.
At least not until you're far more comfortable with the game and it's mechanics.