Here's the drawing without the dotted lines:
I've have to work on the drawings to get the old and the proposed done.
The drain for the first floor shower attaches to the horizontal run to the lav and the kitchen sink so maybe it's wet venting from that. The tub/shower ties into the toilet run and there is no vent unless it's vent from the main stack. The laundry arm is just a wye into the base of the stack. As for the kitchen vent, I can see the vent for it running through the corner base cabinet.
As for what's broken, besides the lack of venting, the is a slope issue with the kitchen drain, I already had to get a section of the cast iron replaced between the first and second floor because it was cracked. The piece of cast iron going through the roof is currently crack.
I'm planning on doing a kitchen and first floor bath remodel and I figure it the perfect time to fix everything.
I'll get that old/new drawing up soon.