The advent are all about late game fleet synrgy, the TEC are about having a strong economy.
Strong economy generally means TEC excel at early game pressure and late game wars of attrition. TEC generally wins late game direct confrontations chiefly when their economy has a large lead so they can field a larger fleet.
In general this means in a late game fleet scenario the TEC fleet isn't supposed to beat the advent fleet in a direct confrontation. Playing to TEC's advantages would rather be trying to divide your opponent's attention by sending border raid w/ ogrovs to multiple systems ata time and once they have divided up(the advent fleet will lose a lot more strength from splitting up then yours as they rely on heavy synergy) their fleet make sure you have enough forces somewhere to cause casualties. Even trade are in your favor- your strong economy means a war of attrition is to your benefit.
Well to be fair I suppose some of this is less true with higher difficulty computers which cheat and gain stupid amounts of resources to work with..
Against computer you can always just build marzas and laugh as their level 6 AoE demolishes the enemy fleet(of course beign wary of anything that can interrupt it- though your titan's disruption matrix can prevent interrupts for 12 of the 20 seconds). Also computers have always been bad at dealing with massed bombers.
That said, the TEC loyalists are a bit UP atm, the TEC rebels are better off though.
As for your specific points:
1). If your weapons aren't scratching their shields there is probably a Progen mothership somewhere on the field- that is your first priority to kill ALWAYS. Make sure to disable it if it starts channeling it's shield restore(that thing swings battles hard). In general bombers are VERY strong for taking out capitalships as well.
2). The Eradica ability that makes it invulnerable only occurs when you kill it- the ship just gets to live a few minutes after dying(the invuln doesn't save the ship, just delays it's death). Moreover that is the eradica's level 6 ability, so any level 1-5 eradica won't have it. In general it's often better to kill the eradica's fleet before trying to kill it. Also, once the eradica goings invulnerable the battle is over- you've killed the enemy titan, time to pack up and leave before it wecks your fleet in it's death throes. This may come as a surprise, but many of the more experienced players think the Eradica's level 6 ability is too weak rather then too strong.
3). The ability reducing all of your ship's health is very likely the Eradica's Chastic Burst as it is the strongest AoE available to the Advent. It's also the main thing that makes the Eradica strong.