Second order is probably the most common passive crossover in most speakers.
Lower order crossovers have a lot of frequency overlap but are preferred for their immediacy & transparency by some. Probably match well with low power valve (audiophile) amplification.
Higher orders have the smallest overlap but soak up power (and possibly dynamics) in the struggle to obtain a very flat frequency response. These would be more common in studio monitor speakers.
Many crossovers (of whatever order) can have extra components added to absorb resonances or to adust phase differences.
Choice is really a matter of taste and the aims of the speaker designer.
There is no absolute best crossover option. Loudspeakers should not be judged on their crossover type (order) or complexity. But only on how good they sound to your own ears.
Yeah, high order x-overs (anything above 1st order, I believe) muck with the phase (timing) of the signal, so for the best phase performance you probably want 1st order or (better yet!) active crossovers.