- Are dragoons mandatory?

No but they might as well be, they contribute the most of any DPS if you include Disembowel and Litany.

- Does mch work with 2 casters and a melee (no drg)?

It'd strictly be worse than having a BRD, really bad. Not recommended.

- Do brds /mch do decent dmg without a drg present?

Definitely the lowest of all classes by default by a small margin without a DRG but not so low you would omit them from your group. You still want/need a TP/MP battery.

- Is dual casters and a brd end game viable?

Sure but worse than 2 melee 1 ranged 1 caster. Not much worse though, casters can do a decent amount of personal DPS with a BRD but considering they don't buff anyone but themselves it's not fantastic total raid DPS.

- Is 3 melee end game viable?

During progression, never. During farm, it depends on the fight, some fights just aren't designed with that many people huddled around a boss mechanically, some are actually quite kind towards as many melee as you want. This raid tier, it's actually a solid comp for A3 and A4 and I think some groups will start using NIN DRG MNK MCH once they've got alts geared, my group included.

- Is war necessary if your party doesn't include a ninja?

WAR's necessary period. Groups using DRK + PLD are gimping themselves. That said if you just straight up will not bring a WAR ... yes, you must bring a NIN, or you will lose a whole lot of DPS.