I have a third bathroom that is only a toilet. I have the vent pipe that runs next to the toilet. I want to remove the toilet and put the washer and dryer in its place.

Question (1) can I use the old toilet drain for the washer drain?
Question (2) Do I need to tap into the vent pipe for the washer to drain?

Question (1) can I use the old toilet drain for the washer drain?
Depends upon how you wish to configure it. Care to share?

(2) Do I need to tap into the vent pipe for the washer to drain?
Again! You're asking a specific question without furnishing any details.
What floor id the toilet on? Anything draining above it? How's the toilet vented? I need all the information you can furnish. Regards, Tom

Tom is asking very specific questions in order to give you the most accurate professional opinion. I agree, more detailed info is needed to provide you with good advice...