Hello all I've been a long time reader of this forum but this is my first post.

The basement of the house I just purchased was rough plumbed for a bathroom. First order of business is to see if my assessment of the pipes are correct. As you can see in the picture I believe:

1-toilet vent (if I was to add a toilet I think I would connect 1 to pipe 4)
2-toilet drain
3-sink drain
4-vent pipe

Does this seem correct?

So the thing I am trying to accomplish is to add a sink where the red box is in the picture. Can I cut into the current waste stack and add a 3in-3in-2in Y (the top green circle in the picture) and run the 2in to the new sink? If so can I then tie the pipe into the vent pipe labeled 4?

I don't plan on doing this work myself, I am just framing and wondering if it's possible so I can frame the spot for cabinets and a sink. I called a plumber and he told me it should be no problem and most likely he wouldn't have to use the vent pipe labeled 4. He said because the sink was located within 4ft of the stack it wouldn't need to be vented. Does this sound true?

Also would it be a good idea, or could I add a backwater valve where the lower green circle is to prevent waste from backing up in the basement sink in case the waste line got a blockage?

Thanks for all your expert help and I look forward to being an active member of the community.

Thanks.