No and yes.
Amendment (not from a professional plumber!): If you were simply running the sink drain into the shower drain and if there was not a reduction in pipe size, I would think you would be able to do that. I think the reason you need a trap for the sink has to do with code that requires a trap be no more than 24" from the basin to the trap arm. That being said, the pedestal sink in my house does not have a trap until after it drains below the floor.
You need at least a 3 in. pipe -- shower is 2 in. pipe
If you reduce the soil connection to 2" and install a trap
typically shower drains are 2"
If you have a drain available. You may need to chip some concrete around the drain to install the bottom half of the drain. With that in place, set the shower over it and the top piece screws into the drain to form a seal. You can build a 2x4 frame and raise the shower up enough to run a drain to a sump or a floor drain.
2"
You need at least a 3 in. pipe -- shower is 2 in. pipe
If you reduce the soil connection to 2" and install a trap
Shower drain goes into a trap under shower. Continue piping from trap to drain line.
Water won't drain from shower. Either trap or shower drain line is clogged.
The toilet is 3 or 4 inch and the shower is 1 1/2 or 2 inch. You could reduce the toilet to 2 inch, but it would never flush correctly or be of much use. The current toilet drain has no trap in that section because one isn't needed. The toilet itself is the trap. If you are using existing waste lines, they should already be connected to the vent stack. If you are adding lines, connecting to the existing stack is fine as long as it is above the last drain.
You can use the existing floor drain lines most likely you will have to cut the drain line or add to it to line up with the hole in the shower. tubs have drains on one end while most showers have it in the middle. You can buy a 30" by five foot base from Mustee. Nice product, comes in left or right hand drain.
YES.
typically shower drains are 2"
A shower rough-in usually refers to the distance from the wall to the center of the shower drain , measured out away from both walls in a shower to the center of the drain. It also maybe referring to the height and the center line of the placement of the shower valve.
The center of the shower floor where the tile shower drain would normally be positioned ... drain base. After this step the sloped mortar bed should be ... pan liner or membrane in our instructions.
Possibly a caulked joint ( lead and oakum )
Usually 8 - 16 " off floor.