You cannot run the shower vent below the floor.
The shower vent and the lav vent must be vertical until they are at least 42" above the floor. Once they are above that point, only then can they run horizontal. (You still have to maintain pitch so that any liquid in the vent runs to the drains.)
They can be connected together and routed up next to the stack to their connection point, which must be at least 6" above the flood rim of the highest fixture connected to the stack.
It might be possible to wet vent the shower through the lav if it is close enough. In any case you have to have a proper dry vent for the lavatory before you can consider it.