Rather than provide an exhaustive list, essentially the same measurements made in the speaker reviews by Amir here in this forum and the speaker reviews by Erin's Audio Corner. If the speakers are active, the same measurements made by Amir for the electronics.
It depends on price, length of the warranty period and the availability of replacement parts. I like an all-in-one solution, but fom a new company, I probably would prefer separates unless the system is inexpensive enough that I don't mind replacing it in 5 years or the electronics are available elsewhere if the company goes out of business (e.g., Hypex plate amplifiers).
Very important. The more expensive the system, the more important it becomes to me to be able to repair it, which may include replacing failed components.
Good spinorama/directivity, low distortion and sufficient SPL are thresholds for me to even consider purchasing a system. Without them, I am not interested in the system, regardless of the architecture. Also, see answers to 2 and 3 above.
Designs that are not rugged (e.g., see the Topping B100 amplifier thread - good measurements, but too many failures in customers' hands), lack of PEQ/room correction features, lack of full featured subwoofer output and speaker/subwoofer integration (selectable crossover frequencies, crossover topologies, crossover slopes, and continuously variable time delay), connectivity issues, and speakers that look more like industrial hardware than home furnishings. The ability to import and use FIR filters is a benefit, but do not forego the ability for the user also to define and set IIR filters.