Some tips: Look for natural chokepoint planets. AI doesn't bypass. Use starbases supported with repair bays and hangars. (And later with fleets, especially against TEC AI that loves Ogrovs.)

Keep them near the star, and know that you can build four starbases at a star. AI doesn't always attack those, and the phase lane destabilization helps a lot.

Honestly, I don't even research the majority of combat ships, so I can't really answer your second question...

Scout. As Kitkun mentioned, by the time you take your first Asteroid (or perhaps one other planet if its a big map) you should find some chokepoints and dash to them right away. If its 1 on 1 (which I recommend at first until you get a decent feel for the game), this will probably become your frontline base. If you have multiple enemies in opposite directions, that's where you starbase up. With a proper setup like Kitkun mentioned, you can keep the AI from taking the planet with a minimal fleet if any at all. Then you can focus taking them out one at a time.

From the very beginning to first contact with the enemy capital in a neutral well you want or a planet you control. The number of planets and how long it lasts depends greatly on the map set up. I'm not particularly skilled at rapid expansion but I can get around six if the map setup cooperates and there is average distance to the enemy homeworld. I usually go in one direction, but few maps put you in a central position so you usually only have two to choose from anyways.

What units you get and when you get them vary greatly both on your strategy and race. I personally rarely get HCs until well into the late game unless the AI has a lot of LRMs and not many strike craft. That said I usually play Advent and can afford to go a heavy capitalship and strikecraft approach against the AI. TEC has much more incentive to get HCs while Vasari has even less.

That said, most players will have three distinct fleet phases. First is your starting capital ship and the rest light frigates as you use up all your starting fleet supply for quick expansion. Second is mid game with a backbone of your basic frigates and level 3 support cruisers, so combination of LRFs, lights, flak and possibly carrier and repair cruisers. Late game you get Heavy Cruisers and other support cruisers/improved abilities, though these are usually added to your mid game fleet instead of replacing them. That said the best fleet is the one that counters your opponents the best, so these are at best guidelines of what to build when.

That kind of situation is why I would recommend playing 1 v 1 for a while. Also make sure you lock teams so that the AI will fight among them selves and not just gang up on you, or do some early game diplomacy so you can get some cease fires if you leave it off.

A proper Starbase would easily handle two Halycons. Even if C sent a big enough fleet to destroy it,  it should buy enough time to get your fleet back. Also you might want to built a frigate factory at that planet or the next one over and try to build a mini fleet to hold them off or even push them back.