(Quoting Ashantai) is there any reason building tiny or small ones anymore?

  Fleets can hold more tiny ships - good with military starbase assistance because each ship gets the defence/attack value added.

  So a +1 ship assistance on a fleet of 4 ships gives that fleet a (free!,  once the starbase has the module) +4 attack.  Ships need more than zero defence to get the starbase defence assist.

  I haven't worked out which ship of a fleet gets attacked first,  but I'd go for that if it is the largest.

Zarnick, I donn't remember the specifics, but the advanced modules act as a modifier on the targeting equation-something on the order of 25 or 50 times normal.  While it's possible to have ships other than the module ship be targeted, it's not likely unless it's intentional.

One additional caveat: Due to the way the targeting algorithm works, if you place no weapons on your module ships, they'll be targeted last.  While not quite logical, this is presumably attempting to emulate the fact that your weaponized ships are trying to protect it, or another similarly unarmed vessel, such as a transport.

To upgrade the asteroid fields you have to research the "Space Mining". Then you have to command your space miner ship to go mining by either clicking on any given asteroid field or by setting it to automate.

The trade thing with starbase is that it has to be in cover with the trade-route lines to give its effect, which I refer to the starbase's area of effect ring thingy. I found it good to put an economic starbase right on the trade lines. As long as the frieghter passes by the starbase's area of effect it gives the trade boost.

Trade goods gives often abilities to your whole civilization (overall). For example the "Harmony Crystals" gives +20% morale which means when you have build it it increases your morale by that percent in your stats. If you have +10% morale in your stats and then manage to get the Harmony Crystal you will have +30% morale, which on every planets gains its benefit.

The tiny and small hull ships can be very good if you got much miniature ability. As it lets you put more stuffs on ships. Now I am refering to the tech "Basic Minaturization". When you get more of does techs the small and tiny hull ships can be pretty useful. As they cost very little but does not let you put much stuffs on them until you have a higher miniaturization skill.

Planets should always be colonized no matter the cost. They can be put to good use later on, even if it is PQ 3! When you later gets the biological techs like "Soil Enhancement" you will understand.

Mandrall

Secret Police Center does not do exactly what the description says; I forget the specifics, but it's effectively useless.  I believe it's 20% for one planet.

Harmony crystals are not bugged*, however you will have to re-adjust your tax rate to notice the approval numbers change, or force the game to update the numbers by opening Civ Manager and viewing your stats.

Incidentally, there does appear to be a bug, wherein trade goods built by another race do not give bonuses when you purchase them in trades-and this applies to selling them to the AI as well.  However, if you build it yourself, that's not an issue, and if I remember correctly, only Gravity Accelerators were so affected.

TA changes things up a bit, yes, with the Yor in particular, with new dynamics for farms and morale, but you still don't want to go over ~20B per planet (with, literally, maxed out morale), or perhaps as much as ~22B per HW.

One of the biggest problems with the alternative trees, IMO, is that they barely have access to any trade goods-which contributes significantly to the morale shortfall you're experiencing.

However, as TA does add a number of race-specific mining and/or production modules, you ought to be able to fill in much of the gap if you manage to secure sufficient morale resources.

You could always ferry the excess population to your less populated planets before demolishing some stalks, using transports.
If you don't have decent morale buildings and technologies yourself, it's often a good deal trading to get some off the other races. Buildings such as a political capital on your homeworld help a lot to get its approval to decent levels (though still lower than your other planets)