First off, the Iconus Guardian has an abysmally low rate of shield regeneration, to the point at which it's a single-use unit unless you have some way to restore shields. It's too expensive to be treated as a single-use unit, so you need a Progenitor Mothership to use it effectively.
You've also gone for drone hosts too early. The problem with carrier cruisers is two-fold; first they are the slowest unit in your fleet by a significant margin, and second they have tight antimatter constraints. If you really feel you need more strike craft presence this early, build carrier capital ships, which do not suffer these issues. You pay a premium per squad, but you get much higher reliability and staying power which matters a lot in the early-game. Given that he grossly outnumbers your strike craft forces, I'd recommend going pure fighter at this point. You can always convert to bombers later, but neutralizing his bombers is your top priority at this time since you're a more defensively-oriented fleet.
Drone hosts work well when you're scaling up to a very large fleet. Huge masses of strike craft have an absolutely terrifying firepower and mobility, but their durability is suspect. You really need to have a critical mass to make them work to their full potential. Vasari bombers have the phase missile upgrade (this is probably what is killing you so badly) that makes them very effective even in smaller quantities, so for small fleets like this you won't get the same oomph as Advent as you would as Vasari.
1. How do you fight carrier based fleets in general? Flak or fighters? Or...?
The telekinetic push ability of the Halcyon Carrier, combined with a fair assortment of fighters, is your best course of action. Don't expect flaks to have much impact against bombers. They'll help, but they're mostly for dealing with fighters and lightly armored frigates. Be sure to manually time telekinetic push; it's a powerful ability but it needs to be used at exactly the right moment.
Beyond that fact, you want to retreat your capital ship pre-emptively. If the enemy strike craft are focusing on it, then have the capital ship disengage and prepare to jump out. Strike craft cannot jump independently and won't be able to pursue. Especially with Advent capital ships, if you start retreating after your shields are already down, it's probably too late to save it.
Your biggest problem is that you're outgunned too heavily for your defensively-oriented fleet to prosper. The Vasari probably has several levels of phase missiles upgrades, which are going to render your Guardians fairly ineffective. He also outnumbers you, has both a fearsome strike craft and frigate component to his fleet, and is fairly well-balanced composure (too much LF, but he outnumbers you enough that this matters little).
Your best option for fleets of this size is to simply compensate with more firepower. Choke him with combat frigates and high damage output. You could probably beat him with massed Disciples (he doesn't really have enough Assailants...) if you just went all-out on that one unit type.
2. When do you start to do military weapon researches like shields, lasers etc.? Is there a rule of thumb? Do I build up my fleet at first? Or am I doing this step by step from early game on (like I did above)?
It's a matter of cost:benefit. The cheap tier 1 techs are worthwhile fairly early, but beyond that technology prices increase rapidly. Given your fleet size and composition, the only ones I'd even consider would be hull upgrades. Beam damage is the only one relevant to you, but T3 is too pricey for your small fleet. You've got more than enough shield upgrades already if you have guardians, so really hull is the only one I'd go for.
You need a bigger fleet more than anything, so I'd avoid tech for now.
For the record, Advent vs Vasari is by far the hardest matchup in the game. It's actually quite masochistic in multiplayer. You really need to go all-out on your firepower and out-gun the enemy just to even the scales.
