Piano and dining set: been there, done that. Unless you are 100% certain you will be in China for over 4-5 years, don't ship the piano. Yes, pianos are shipped all the time internationally, but properly crating and protecting the piano before putting it in the container isn't cheap, and you still risk damage, and you'll have to find a tuner when it gets to the destination. China has some surprisingly decent pianos--which you can buy and then resell or just rent. Friends of mine (musicians) bought a new Pearl River upright for about RMB 12000, then decided to move back to USA a year later and resold it for RMB 10,500. (they already had 2 pianos back in USA so didn't want to ship back). Or pick up a good used piano from another expat leaving China. Beijing in all seasons except summer will be drier than optimal to keep a piano in proper condition. The summer is not the season you have to worry about, it's the lack of humidity at other times of the year. Beijing dryness plays havoc with wood products (they tend to crack and split).

As to dining room, I wouldn't ship--buy in China. I've shopped for this before and you can definitely get a nice 5-piece dining set for RMB 6000 or less. RMB 7200 will give you more options. South 4th Ring Road furniture market (or smaller version on north 4th Ring) is great place to start. This will be miles better quality and range of styling than IKEA, whose merchandise I don't care for. I don't know where you are moving to in Beijing, but most places come furnished, so chances are it will already have a dining set (which you can negotiate with landlord to either move it out or replace it with something more to your taste.)

If you need to fill your container, make books, professional, and personal items such as clothes/shoes be your priority.