1.Trade ports are definitely viable, especially once you get 5-6 planets. Until then, don't get them, you don't need them yet.
Culture is good if you have a small clustered empire. The thing about this though is that it boosts allegiance, and thus income so the more you can affect with one, the better.
Refineries are weird. There are some situations that scream refineries, but most of the time trade ports are better.
Passive income increases are good. Since you are playing against the computer, this is a good idea, but if you ever play online, you may not want to get these as you will be putting your money in a long term benefit rather than a short term one to counter the rush that you know will come.
2. Heavy Cruisers (HC's) are good. They should constitute a good chunk of your fleet late game and act as an excellent meat shield and deal good damage. These ships counter just about everything.
Long Range Frigates (LRF) are great. These are generally the primary damage dealers in the fleet. If playing as Advent, get illuminators. They are great and also remember that they have guns on their sides as well. These ships counter LF especially and do pretty well against everything else.
Light Frigates (LF) are decent. They used to be worthless as they would just get slaughtered by LRF, but now they have been buffed. These counter cruisers, especially those with little damage capacity.
Carriers are good. They can use Fighters or Bombers which go after light (LRF/bombers) and very-heavy (structures, HC's, capitals) targets respectively.
Also, you should only get upgrades when it is cheaper to increase your overall stats via upgrades than by buying more ships (or if you have hit the fleet capacity limit and don't want to increase it due to the drain on your economy yet). Generally, more ships is good. Don't get carriers too early though. They are expensive and you won't have the funds to deploy them.