There are three main mistakes new(ish) players make in Sins IMO. The first is building too many capitalships. The second is researching too much, especially early. The third is not expanding fast enough. I'm guessing you're falling victim to 2 & 3.

First of all, you should never get mining techs until you have a good sized Empire, and in fact I go through the vast majority of my games without getting them at all. Terran pop upgrades should only be gotten if you can't build any more ships and don't want to expand your fleet at the moment. Rest are all good techs you should have in the first half hour (Well the neruda depends on how face your diplomacy is going, if you're under serious attack put off diplomacy until later), but you should replace hangars with repair bays.

For the first phase of the game, most of your resources should be geared to expansion. Get Corvettes ASAP for your initial expansion push, and if you have time to get ice/volcanics get those colonization techs. Timing is key with the frigates, the only important thing with them is that you have the ability to build them when you need that type of frigate to counter you enemy. If your enemy is spamming LRFs for example, you really have no reason to build anything besides corvettes.

That is indeed a big problem. You may want to try "double expansion". Most pro players will open with a colony cap and an additional colony frigate or two colony frigates so they can expand in two directions at a time. Your colony caps can take most volcanic/ice or lower militia without support, so that frees up your Corvettes/LF to clear the asteroids & dwarfs, and possibly weak volcanic/ices at the same time.

Also make sure you focus on destroying the siege frigates first. Once all of them are destroyed, you should colonize the planet immediately. Even if you're forced to retreat, the militia can't retake the planet from you, so you're free to get the planet pop upgrades.

Also, while Gemini is probably too small for this, keep in mind as TEC Rebels you have Truce Amongst Rouges (TaR) at your disposal. Rushing this tech when you have some breathing room lets your colony frigates colonize without armed escorts, and lets you bypass militia defenses without issue, freeing your fleet to raid the enemy while you colonize most of the map. This takes a lot of practice to do this well though, and you'll end up spending a lot of cash getting basic population upgrades and extractors, leaving you vulnerable for a bit until those planets repair their development costs.

Early on, you'll just get the first/second population upgrades for asteroids & dwarfs/other planets and the resource extractors. Build research structures as needed, put unless you're pulling of a strategy like TaR you really can get away with only two of each lab early on. As TEC your early trade is a big advantage, but keep in mind you need have the military force to protect that trade, and that you should probably not invest in trade until there are no more neutral planets you can take (planets pay for themselves faster than trade ports). Only once all the easy planets are taking should you invest a lot in trade, and only if your Empire is secure enough.

Also another mistake new players tend to make: Be very careful of wasting too much money of static defenses.

Turrets are practically only used to speed expansion(colonize a planet before killing some of the defending militia and build turrets immediately to help kill the defenders faster, then scrap the turrets once they finish their job).  Hanger bays are luxury late game defenses for when you have a ton of extra money.

In general 2 repair bays are enough to defend a planet as long as there is a frigate factory or two somewhere relatively nearby.

Likewise don't overbuild starbases- starbases should be built where you absolutely need them, not on every planet.

Early game a vast majority of your income should be going to either improving your economy or increasing the size of your fleet(build more frigates), dependign on whether you get involved in early battles or are playing for the long run.

Here's a question, when do u start researching structural subsystems & ballistics/rocketry?

You need to have a decent sized fleet before you invest in stat research, otherwise its much more cost effective to just build more ships. Again like the terran population upgrade, if you are at your fleet supply limit but don't want to expand your fleet because of the increased upkeep, you might start getting the tier 1 upgrades after the first or second fleet supply upgrade. After that it just depends, but I will say in competitive matches its pretty rare you max out your stat improvement bonuses. In short, if you have the choice between more stat research and something else, I would go with something else until your fleet gets fairly large.

The main exceptions to this are phase missiles, which as Vasari you should get ASAP, or research that you need as a prereq for a unit or structure, like the missile techs for the Novalith cannon as TEC Rebels.

Probably means you weren't buying the population planetary improvements for the planets you colonized- or alternatively that the AI are just colonizing more planets.  Leaving a planet with absolutely no population upgrades will result in the planet actually having a negative income though- so having a couple planets that you never upgrade the population compacity of can result in a hit to your economy.

Income from planets is just a measure of how high the population of your worlds though, so the only factors are how many planets you own, how many you've upgraded the population capacity on, and how long the planets have had to generate population.