Depends on the duration of your exchange study at EdUHK. If you only stay in Hong Kong for one semester (i.e. less than 180 days), you don’t need to apply for a HKID.
Yes. International and exchange students can obtain medical services at the Health Centre at the same nominal rates as other full-time students. Check out more details here.
Under Immigration Ordinance, Cap 115, Laws of Hong Kong, exchange students studying at Hong Kong tertiary institutions cannot take up any part-time or full-time employment, whether it is paid or unpaid.
Yes, you must apply for a Mainland China visa in Hong Kong in advance. You can apply for this at any one of the China Travel Service (HK) branches. More information is available here. Alternatively, you may apply for a Mainland China visa from the Chinese embassy or consulate-general in your home country.
No, but accommodation options are available via the Hong Kong Tourism Board website.