When we "do the laundry", the laundry must be washed with a washing machine or can be hand washed? Does the phrase has anything to do with the way the clothes are washed? What are some alternative expressions to this phrase? Thanks very much.
Any method of doing the laundry is OK. It's still "doing the laundry"!
"wash clothes" is the most usual alternate.

What are you doing today?
Nothing interesting. I have to [do the laundry / wash clothes].

In Uk we usually say either

Do the washing or Do some washing - we know what we mean without mentioning clothes.
Students: Are you brave enough to let our tutors analyse your pronunciation?
Come to think of it, we do also say "do the wash" (no "ing", though - we know what we mean without adding the "ing"!!! Emotion: smile )
Teachers: We supply a list of EFL job vacancies