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.