so I had about 6 or 7 of them, but all below level 6.
If none of your caps were above level 6, then you definitely had too many. As TEC, I strongly recommend not going above 3 capital ships until at least one is level 6. The TEC capital ships are great if you can get them to level 6, but are actually pretty weak until you get there. You're much better off with a few high-level capital ships than many low-level ones.
1. How was the Vasari able to re-build ships (it seemed) with so few planets and presumably not much cash or resources?
I'd need to see the replay. It could be that he had a lot of money saved up that he hadn't been able to spend yet. Or it could be that you just weren't dealing very many casualties. He also might have been using returning armada to create units at a substantial discount. I just can't say.
2. What are the abilities that the Vasari caps were probably using that made them so tough for my fleets to counter, and what kind of fleet composition/strategy works best for Tec against a fleet consisting primarily of Vasari caps level 6 and above?
The most likely culprit isn't the capital ships, but rather the smaller support cruisers. The Stikulus Subverter can shut down your units, taking their engines and weapons offline. This can deprive you of the majority of your firepower and make your fleet easy prey. Kodiak heavy cruisers are particularly vulnerable to this. On the other hand, the Serevun Overseer can instantly restore hull to a target unit, which can make focusing on a capital ship very counter-productive. With much of your firepower disabled and the remainder getting soaked up by healing support cruisers, you might not kill anything.
As for the capital ships, they bring their own special abilities. The Kortul can regenerate its own shields, adding even more durability on top of what the Overseer offers. As well, it can shut down strike craft weapons in a radius around it, so if you were relying on carriers that can completely gimp you. The Skirantra can heal nearby units in an area of effect around itself, and this healing stacks with the Overseer. The Jarrasul, Antorak, and Vulkoras all have their strong points, but are not going to be a big deal in a straight fleet battle like the Kortul and Skirantra can.
3. There were some Vasari non-cap ships -- should I have focus-fired on those first?
Yes. Focusing on his capital ships first was probably a big mistake. Remember that capital ships have relatively poor damage output, so you actually aren't cutting their damage significantly by knocking out a capital ship. In the meantime, the enemy probably was busy killing your damage-dealing units, so even if you did take down the capital ship, you'd still lose the battle overall.
Be thankful you're not up against a human opponent. There are horror stories of beginners going up against a small force of 10-20 Vasari units and losing hundreds of frigates before they finally take them down. In the hands of an expert, Vasari is downright scary.