The words "partiality" and "favoritism" mean basically the same thing. Paul is emphasizing that Timothy must judge honestly and be fair to everyone. AT: "to keep these rules without being partial or showing favor to anyone" (See: [[rc://en/ta/man/translate/figs-doublet]])
The placing of hands was a ceremony in which one or more church leaders would place their hands on people and pray that God would enable those people to serve the church in a way that would please God. Timothy was to wait until the person had shown good character for a long time before officially setting that person apart to serve the Christian community.
Paul speaks of someone's sin as if it were an object that could be shared with others. AT: "do not join in another person's sin" or "do not participate when another person sins" (See: [[rc://en/ta/man/translate/figs-metaphor]])
Possible meanings are 1) if Timothy chose someone who was guilty of sin to be a church worker, God would hold Timothy responsible for that person's sin or 2) Timothy should not commit sins he saw others committing.