carry-on luggage dimensions and other related

1) Are the carry-on dimensions limits rigid limits in each dimension or limts on the sum? i.e., your website says china's carryon limits are 20cm x 40 cm x 55cm (which adds to 115cm). I have carry on that is 23cm x 36cm x 56cm (which also adds to 115cm) It's hard sided so it can't be "squished". Would it be OK on a domestic China flight?
2) US flights allow 1 "personal item" in addition to a carry-on (e.g., purse, etc). Do domestic flights in China allow this?