There's a Club lounge at Sheraton Dameisha Resort. But there don't have afternoon tea. Only cocktail happy hours in the evening. Many alcohol, wine & beer selection plus lots of snack foods. Only to pay a small fee to upgrade the stay to the club floor.

I have a chinese visa but you can get a short term visa at the border but expect a delayed process

From Hong Kong airport faster way is to take a MPV transport from the mainland transport counter. Cost is HK$150 per pax to Huang Gang border and took only around 40 minutes. After the Huang Gang border China immigration clearance, may take a taxi from the taxi stand to any part of Shenzhen.

I don't think it is private; i.e. fenced off but the hotel is by far the main source of users and is not crowded. Sand was clean when I stayed in December 2018