Basement Bathroom from Scratch

Perhaps you've all read my previous thread about perhaps craming a bathroom under the stairs... after some measuring, it's just not going to fit...

So I figure I can put it at the bottom of the stairs... please forgive the picture as it is not to scale... there actually is enough room to build a bathroom down there...

Anyway.... the toilet to the main line is going to be really close... like 16"!

I also need to plumb in a sink as well... not enough room for a shower...

In red I have drawn where I'd like to place the toilet... please veto me if I'm an idiot... or there is an easier way...
Three questions..

a) where would I run the toilet vent off of with what fittings... I know I cant vent from below by code. I can use an AAV and there is no access to the roof for a vent.

b) How and where am I going to plumb in a sink

c) can I tie the toilet and sink vent together to use one AAV

Thanks,
Jay


Here is a layout...