Can I apply for different types of visas at the same time?

I am currently in Singapore, UK passport holder. I have a new job in Shanghai which is due to start in August, however, I need to enter the country at the end of June as I have to settle my pets before I fly to the UK for a holiday in July.

1. I will only be able to enter the country in June on a L type, which I have confirmed with a relocation company is acceptable for purposes of importing pets.
2. Should my Foreign Expert Certificate arrive before I apply for my L type, will I be able to go to the center and apply for a) L type b) Z type- for entry in August c)letter instructing immigration officials to stamp the L type for my June entry and the Z type for my August entry or
3. Will I only be able to apply for my L type, and then have to apply for my Z type when I am in the UK?