Kahit ano gawin mo, one way or the other you'll need a crossover as you need to set your sound frequency appropriate to your driver. Last thing you want is a tweeter doing full range
Basically, not using the passive crossover that comes with the set gives you
FLEXIBILITY in tuning as you're running a fully-active system. You can fiddle around with it rather than having the passive crossover.
But, the problem with this is that unless you're an audiophile, going full-active isn't ideal due to channel constraints.
In terms of sound quality, most are negligible. In most 2-way systems, the tweeter only needs minimal power to run. Throw around 20W at it and most tweets are happy. Not everybody can afford extra 2-channels. In semi-active 3-ways, most crossovers are for the midrange-tweet while the midbass is active.
Answers:
(1) You'll have a crossover unless you're running full-range speakers. As for passive crossovers pros and cons, look at above.
(2) No. Crossovers use is to simply filter sound frequencies. Think of them as Quality Control: "Oh ito, high frequency... doon ka sa tweeter. Ito low frequency, doon ka sa subwoofer. Etc."
(3) If you have a crossover, use it. Most crossovers that come provided in the component sets are made for that set naman eh. If you have it, use it. If you don't, that's fine as well but check since you need 2 additional channels (for the tweeters).
Good quality crossovers are expensive. So kung meron ka na, go for it. How expensive can they get? If you check Karty.. yun Monty ni Drivensteven who posted recently dito sa section. His fellow Team DSD member, Martin, has custom crossovers in the 6 digit figures. They're freakin' HUGE. Pagpinagsama mo yun dalawa probably magsinglaki lang ng 22" monitor. Or even bigger.