Do I need to get any approval document for "China 144-hour visa exemption" before the trip?

You're eligible for the 144h twov when your transit time in Beijing is over 24 hours.
If it's under 24 hours, you're covered by the 24h twov.

When you fly on one ticket, you don't need any documents at the check-in.
When you have seperate tickets, you should show your e-ticket at the check-in in San Francisco to let them know that you're just in transit in China.