I prefer Oct 7 when the weather will be mild in China. Mor important that you will avoid travel peak week of Oct 1~7.

It will be ok to wear shorts in Shanghai. A thin long-sleeved shirt should be prepare for coming to Beijing and Xian. These two cities do have cool morning and night. There is no special requirement for dinner at night, just comfortable. Maybe a little nice for a party!

You'd better start your journey on 7th, Oct. Since from 22nd on, Chinese People will start their Mid-Autumn Festival Holiday and thereafter the National Day Holiday, every tourist site would be crowded. Anyway, the holiday would finish at around 7th, Oct.

For clothing, it might still be hot in day time, but take a coat for evenings, people don't require too much on what to wear at night for dinners here, but if you are going to some business occasions, just wear as you are in your country.

Must sees here are the Great Wall and Forbidden City in Beijing, Terro Cotta Warriors in Xi'an. the Bond in Shanghai, do some search of these places and you'll find more.
Good Luck!