You will not find an English speaking pharmacist - I haven't in my nine years here. You can get scripts filled in HK but they probably have to be originated by a doctor in HK - so stock up before you arrive. For details of health care see this shekou.biz/medical.html
Be very careful when dealing with the shenzhenrent.com folk as their contracts often push the onus for repairs onto the tenant.
I have lived here 8 months in Nanshan. The HOME hospital near Walmart has a western pharmacy, so does CanAm. I don't know the HOME hospital hours but CanAm are only typical business hours which is nice if you only have emergencies during working hours
The HOME hospital is CLEAN, CLEAN, CLEAN! I called and they set me up with a translator who spoke English and stayed with me for my whole visit. It costs 100 RMB for the visit and was pretty great care considering the alternative at a Chinese hospital. I also tried a Chinese hospital, bringing my dear Chinese friend along. I saw the line of about 200 people and said NO way! Also try wikishenzhen.com They have directions in Chinese and English so Taxi drivers can know where to go.
Good luck moving! Louho is nice Futian has more options for Westerners, though Shekou has the most.