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For the power adapter just check the energetic plate. You'll see input and output there. As long as the power adapter is 5V@1500mah or 5V@1,5A you're good. I actually use 5V@2A for faster charging. You shouldn't go above 2,1a or 2100mah or you'll kill your battery. Slowly it will just get worse and worse battery life. Z1 should limit the current, but I would keep it between 1,5-2,1A anyway.
I had been charging my Z1 with a lots of cables and every one of them was good. Just make sure it's not too long like 5m. (electrical resistance matters)
There's also a module in the stock rom that limits the charging current if your device is running hot. The Z1 can also use more power than it is being charged with. I had this when I was watching a movie with a terrible reception. I dealt with it by turning on the flight mode.