May be getting the hang of an accelerated rush... Super Breeder with Small No-Engine CSs AND a base speed increase seems feasible.

In DA, I'd only have to research for about the first 6-8 turns.  In TA, it could be as much as twice that (12-16), dependent on tech tree.  That's assuming Tough, anyway-things actually get a little easier at Suicidal due to the research difficulty bonus multiplier, but let's not get into that; you need to get Tough down before you move back up again.  (Same goes for Maso, but not to the same extent.)

Breeder is definitely the way to go.  I'd say the only really viable rush strategies are, still, Breeder and Hive.  Hive lets you kick out a 250M pop colony ship wherever you go, but Breeder lets you fill it, so it's a toss up.  Don't forget to stack pop growth bonuses, especially with Breeder-you'd be surprised how much it helps.

1) The 1 colony ship per turn approach assumes it's a high cost, high speed, high capacity, high everything, from your HW.  In DA this would run me about 165/175+.  It's a personal preference between a fast 250M and a slow 750M.  We've even had debates on 500M ships.  Generally speaking, the 250M will net you more planets in less time, but cost you monre money in doing so, due to lower tax income.  In TA it might be a little higher, can't remember why.  I think it just seems that way because it's harder to get that much production in TA.  I've been playing DA more as of late since TA has a reduced planet count.

No-engine smalls works as well, as you have discovered.  I don't want to take you away from that track just yet, if it's working for you.

2) If your tree has it, recruiting center is good.  Fertility clinics are also a good idea if you have them early enough.  But aside from that, yes, it's better to leave it building nothing until you're making back enough taxes to break even with regards to maintenance.  (I don't, but it's better to.)  Generally this should be between 1B and 1.5B, dependent on your tax rate and econ bonuses, from memory.

3) Perhaps the first 20.  If you haven't stabilized by then, especially with tech trading, then you might have a problem.  And by stabilized, I mean, not in danger of going red-it doesn't necessarily mean you make more than you spend from taxes; just that you can get it from the AIs or anomalies, or a combination thereof.

4) I've found that your actual diplomacy ability doesn't make that large of a difference, so long as it's somewhat greater than the AI's (and since they don't research yellow techs very often, as long as you're not against an SD, you should be okay with Interstellar Republic or Star Democracy), but you will want to pick up Galactic Bazaar, if you can afford to.

I generally don't do a lot of tech trading in my colonization games, to be honest-just enough to keep me afloat.  Historically, anomalies have carried the day.  And in answer to your question, as of DA 1.7 or 1.8 something, they do regenerate; they're just a little slow.

5) I like 6-8 survey ships..  Smalls are good, tinies work as well.  They cost less, speed is not overly relevant, and they pick up the HP bonuses from Starship Graveyards and can later be turned into fighters that can take quite a bit more punishment than their peers.

6) TA IS a little harder to manage, especially with focus dialed down to 80% of its former value (20% given instead of 25%, but 25% is still lost).  I always get planetary improvements as soon as possible, if only for the +10 research.  Xeno economics early helps out as well, although you're probably losing so much money you almost don't notice.

Gmoonii seems to assume you're playing SD; that's the only SA that lets you speak to AIs every 4 weeks.  Everyone else has to wait 8 weeks.*

For Breeder, you either want 100% approval, or 41% approval (or, if you're desperate, 21% approval).  There's no point being anywhere in between.  And that's a minimum, on a planetary basis, so your empire-wide might deviate from it slightly.  I'll typically run 100% only when I get my research done so that I can pump out a colony ship a turn without losing pop from my HW.  You can also consider building a colony ship a turn on your secondary, if you can get enough production built up on it, and using the pop from your HW for it.  Generally speaking, 41% is sufficient for non-Breeder.

Apologies if any of this is not quite 100% on the ball; TA is not my game of choice, although it's still worth the money I paid for it, and besides that, although I miss colonization and will go back to it some day, it's not my preferred strategy at the moment, so I'm taking a lot of this from memory.

My primary tech rush is 0/0/100, with a xeno lab on a +300% tile.  For a colonization game, I'll switch over to 100/0/0 at 6-8 turns in.  I generally rush buy my factories (I don't need many) and set focus to research.  TA's focus nerf and general lower production values make it more difficult, but it can still be done.

I don't imagine a ship every 8 turns is helping you at all.  You might consider dropping a single factory on a new planet if you intend to colonize from it-it makes all the difference in the world.  With that AND adjusting your sliders, you might be able to get it down to 2 turns, which is worthwhile.  It may cost you a little more, but you'll get the planets sooner.

If you're coming from DL, DA is probably simpler; at least you don't have to worry about the different trees.  One thing I don't like about TA is the change to the tech capital to function as a planetary racial bonus rather than as a doubling multiplier, but I'm kind of rare in even knowing about things like that, much less being bothered by it.

I actually liked the Torian tree for colonizing-you get the Central Mine, at least.

I spent probably 6 months on DL even after DA came out, but eventually Breeder and Hive won me over.  I was in TA's beta, but TA still feels like a beta to me (although it is enjoyable), and I'm now going through with DA what I went through with DL-not wanting to give up what I know and like.

If you haven't colonized your secondary on turn 1, especially with +speed, there is something very, very wrong.  You also want to get a third planet up ASAP (with your miner, convert it into a colony ship), since you can only research x+1 techs per turn, where x is your planet count.  (This is equal to x rather than x+1 in DL, by the way.)

It might be worth it to set up your asteroid mines for a colonization game, but I can't remember needing to.

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It's possible that your colonization approach, or any that I recommend, may cause you to go into debt, but it shouldn't be for more than about 8 turns, and at some point, you'll get the hang of it and be able to do it while spending between 0 and 2 turns below -500BC.  But initially, you're going to see a lot of your strategies send you into unrecoverable debt for over a year...it's how we learn.

Social focus should work fine if you're no longer building ships on the planet in question.  Even in TA, military <-> social transfers 50% base, rather than 25% (20%).

I already suggested you plant a factory on any planet that you intend to build things with.

Glad to hear your tech rush is better.

I'd be more concerned about lowering approval to max population for a higher base tax rate than market centers, though.  Especially running Breeder.

Oh, and governments help as well.  If you can manage to get yourself to Star Federation, it actually gives a 40% bonus, even though the game says it gives 30%.  Governments are important because they're a multiplier-they don't add on to your other economic bonuses.  So if I have a 40% econ bonus and a 20% government bonus (Star Democracy), instead of 60% I have 1.4*1.2=68%.  The more bonus you get, the better its effect is.

I'll add my 2 cents:

1) Try to get some technologies with an econ, morale, production bonus from the AI, purely by trading away influence points. With the tech trees being different by race in ToA, you can get bonuses from technologies that are not available to you through research.

2) When setting your taxes in function of approval, check the list of your planets: it's not necessary to have 100% or 76% average approval since the population growth is determined per planet. I often find that I get a lot more income (or rather, less negative income) by having a few planets below such an approval threshold, as long as the majority of my planets still gets a pop growth bonus.

3) Trade ships. I really depend on those early on in ToA to keep my economy afloat. While it's not best practice and more experienced players will find way to avoid this, I find that it helps if you need the additional income. In addition it improves your relations with the AI, so choose wisely with whom you want to trade.

4) A mistake I often made, was to go for the most distant planets I could grab (at least up to medium or large galaxies), in an attempt to get a big span of control early on. In fact, it's better to choose your first few planets to colonize closer to your home planet: the sooner your colonies are up, the sooner you can have them produce more ships.