In the process of building a garage addition that has living space above. There is a single bathroom on the second story with a lav, toilet and shower. The bathoom is 12' long with the toilet and shower on opposite ends. The main stack will pass up through the wall next to the shower.

Just getting ready to start my rough in and wanted to run my dmv diagram past everyone here. My main questions are:

1. First off, does this diagram look correct or am I way off?
2. There is a utility sink on the first floor, I know I can't vent it using the bathroom lav drain but can it connect to the lav vent in the attic? And can that vent tie back into the main stack or does it need its own roof penetration? In my drawing I have it going up through the roof.
3. My toilet is roughly 12' from the main stack so I've added a vent. Would a 2" vent be sufficient for the toilet?

Any help/feedback is appreciated.