Let me put it this way:
1) When you are a 1L or LLM student, the firm will look at your 1L GPA/Experience/Interview skills first, then look at your visa status second. Your employment will be conditioned on the approval of your H1-B visa. HR handles the process without much intervention from partners.
2) When you are working in a law firm, whether the firm sponsors your green card application depends on your performance and recommendation from at least one partner.
Very few China-born attorneys have obtained green card while working in law firms. I do not know what happened to those who went in-house instead of staying in law firms. In theory, they can file green card applications as well.