Should i first go for Military Research Labs or Civilian Research Labs, if i want to setup a strong economy quickly?
If you are Advent or TEC, always get at least one military lab early on for Corvettes; they are fast and good against siege frigates so they are good for clearing out neutral planets and are a powerful combo with Light Frigates early game. After I usually go for 2 civic labs for planet exploration techs, but as TEC this will also get you trade ports. Advent will need one more lab for trade.
How soon should i get the culture spreading buildings, if i am going for a strong economy from the start?
I wouldn't rush specifically to get them, but if you end up getting enough civic labs for them while trying to get other things and have some spare resources to invest, go and put one culture center down in a central location in your Empire. Culture doesn't contribute to your economy as much as trade but it has other bonuses.
In pure economic utility, Trade Ports > Culture > Refineries.
Is there any benefit to making multiple Trading Ports at every planet?
Sure, a planet with three tradeports makes 3 times as much credits as a planet with one trade port. Late game you will have many planets with lots of trade ports. However, even better than building a trade port at an existing planet is to build a trade port at a new planet that expands your trade chain. This will increase the trade income of all trade ports in your empire, as well as giving you the money from the new trade port.
How often and how many Orbital Refineries should i build?
Sadly if you just play the base game refineries are rarely any good. You need a centrally located planet with many phase lane connection to planets with lots of resource extractors (preferably non-colonizeable gravity wells) to make refineries work. On large maps especially, where you can get very long trade chains, trade ports are almost always better. You would not be a bad Sins player if you just always go for trade instead of refineries, since there are relatively few occasions where refineries are the better deal. It is usually more profitable and simpler to ignore them and use trade profits to buy resources on the black market (especially for TEC who can research a discount).
In what order should i develop/upgrade/build a newly colonized planet?
Get 1 (asteroids/dwarf)/2 (Everything else) population upgrades. Once you have money to invest, build logistical structures (Research, trade, culture etc). Then invest in more logistical slots, and build more logistical structures, until full.
All the other upgrades are situation. Planet Health and tactical slots are obviously only needed for planets in danger of attack. Exploration is usually a luxury for those with money to spare, but if you do get that huge economy why not.
Do i first make the metal/crystal extractors, or the Trade Ports/Refineries, or the various planetary upgrades?
Extractors! They're cheap, pay themselves off quickly, and don't use any planet slots. When first colonizing a planet the population upgrades and extractors are what you should always build right away without question.
With TEC i find it easy to get a strong economy quickly running, but with Advent and Vassari i am having major issues.
Yeah, the TEC are all about lots of simple and efficient ships. The Advent and Vasari playstyles require more finesse and rely on using powerful abilities or research well, not as much on numbers and brute force.
About Fighters and Bombers, in what amounts do i actually need them?
I tried multiple combinations (5 fighters 5 bombers, max bombers, max fighters...) and they all seem borderline useless?
My Fighters/Bombers attack the enemy, and i lose all of those squads very quickly.
How do i utilize strike craft vessels?
How do i make strike craft actually do something, instead of quickly dying at the start of the battle?
Are those anti-strike craft ships, like the Defense Vessel, actually useful?
You don't have enough fighters and bombers. Strikecraft are most deadly when you have a critical mass of dozens if not over 100 squadrons. Since strikecraft do not have collision detection (they can fly through each other), they can attack any single unit at once, while antifigter units (including other fighters) will have difficulty efficiently targeting that many units. While your strikecraft may eventually be destroyed if the enemy has lots of antifighter units or ships with abilities like Flak Burst, it is nearly impossible to protect key important capitalships, starbases or even titans from a properly executed bomber swarm.
Defense vessels aren't bad, but unfortunately they are mainly for fighters and corvettes. They do not do as well against bombers, which is what you usually fight more of late game since they can take out starbases.