Crossovers

High order is needed when drivers are used to the extremes of their operating range. Often high order notch filters are used with modern light weight metal or ceramic drivers - this is an attempt to reduce the out of band "ringing" - it often works although at higher output levels ringing can still be audible issue.

High order filters and notch filters may introduce phase shifts that can be detrimental, however, they seem to be common these days. In the past designers would stick to first order filters.

Most Coincident speakers use 1st order and are happy with 3-10 watts due to 94-102db efficiency.

The answer is most likely yes. A first order XO on a tweeter limits the low frequency signal sent to the tweeter, below the crossover point, at the rate of 6db per octave. Second order is 12 db rate, third is 18db rate, and forth is 24db rate of cutoff. If your crossover point is well within the operating range of your tweeter power is not a problem. So if you need to use the full operating range of your tweeter, as in a two-way speaker, a higher order might be better. From all I've read there are pros and cons to every type and order of crossover. I'm partial to third order, but don't ask me to explain why.

Think of crossover parts like speed bumps, the less you have in the speaker's crossover, the easier it is for the partnering amplifier to drive the loudspeaker as there is less for the amp to push through.

Therefore, next to speakers which use no crossover (Fostex, Lowthers, AER, etc.) a first order crossover is the most low power amplifier friendly.