144 h visa with trip LAX ----> HK ----->shenzhen.-----> HK---------> LAX

Sorry to say that your route is not eligible for the 144. Flight route like LAX - HK - Shenzhen - LAX is an eligible one. Besides, you cannot take train from HK to Shenzhen to enjoy the 144, only Shenzhen Bao'an Airport can issue the 144.

An alternative: You can take a ferry to Shekou Port or take MTR to Luo Hu (Lo Wu) Port from Hong Kong and apply for a Shenzhen Visa on Arrival at the visa office.

Ok. I can switch to the following itinerary. LAX----> HK----->. Shenzhen ---->LAX. I will stay a few days in HK first, I will then take the MTR to Lo Wu and request a 144 visa to Shenzen. Will stay up to 6 days and go back to HK. Do I need to buy the transfer back from Shenzhen to HK ahead of time to show to the visa office, or just my return flight from HK is goo enough.