There is no simple answer to your questions. Mostly it depends on what techs the enemy is employing or not. Usually a particular weapon combo that works well against one enemy might not work well against another. SO you need to redesign your ships between wars in anticipation of the enemy weapons systems you expect to see in the coming battles. What works in one game doesn't necessarily apply every time.

As long as the enemy doesn't have Class-V shields yet, you can usually get by with any of the beam techs, although some work MUCH better than others against certain defensive tech configurations. The most effective designs of course take advantage of miniaturization and any available HV AF CO NR ENV AP upgrades. Personally I like AF Ion Cannons on a glass-cannon design or Neutron Blaster/Tractor Beam combo on a tanky ship design.

You probably should check out Gontzol's Weapons Calculation FAQ to see how the math works for the various weapons systems. That can help you decide whether it is worth improving your own fleet's accuracy or evasion techs.

If you are having problems damaging enemy ships with fusion beams, then there is no point in switching to fighters until you get Phasor tech.

The "best" mid-game anti-missile tech is probably Lightning Field, since it always activates and destroys 50% of incoming missiles, regardless of missile type. But if you have good targeting computers, a ship armed with a very large number of beam weapons can be effective at shooting down missiles too. AP doubles damage against missiles, and ENV quadruples damage.

Having one or more dedicated point defense ships that protect the rest of the fleet from any incoming missiles is definitely an option to explore, if that fits with your battle strategy. But the AI will often target the weakest link, so don't count on your point defense ships being missile magnets. They will likely have to aggressively position themselves so as to intercept missiles targeting other ships.