[LazyPup]

You stated that your toilet will need to be on a step. In fact it will require neary two steps upward:
1. The existing drain line is 1" above the basement floor.
2. You will need approximately 12" vertical clearance to allow for the closet Bend or 1/4bend (90 elbows) to transition from horizontal to vertical up to the closet flange, which is at floor level under the water closet.
3. The run is 10' and you must allow 1/4" per foot pitch- 10 x .25= 2.5"

Thus your total rise will be 1" + 12" + 2.5" = 15.5". Allowing that a step commonly has an 8" rise, this will require two steps up to the platform where the toilet is installed. The concern then would be how much ceiling clearance you would have.

While there are hundreds of different Plumbing Codes around the country, they are all primarily based upon either the International Residential code (IRC) or the Uniform Plumbing Code(UPC).

The IRC and UPC take some very differing views on the issue of venting.

Under the UPC all vents must rise vertically and must terminate through the roof. Flat horizontal vents are prohibited, and when it is necessary to make an offset in the vertical riser, the miximum angle is 45degees from vertical. (The UPC will permit a maximum of one Air Admittance Valve -AAV or Studor vent per structure, with the written approval of the local AHJ (Authority Having Jursidiction).

Under the IRC all structures must have one "Main Vent" that runs undiminished in size from the building "Main Drain" up and out through the roof. Additional auxillary vents may terminate through the roof, terminate in the attic space, terminate through a side wall (Many restrictions on location of sidewall vents), unlimited use of AAV's.

In all cases a water closet (toilet) must be located on the end of a horizontal waste arm. The UPC limits the length of the waste arm from the water closet to the vent at 6'.

The IRC permits an unlimited length of the water closet waste arm.

If you are under the IRC the simple solution would be to install the water closet on the end of a horizontal waste arm. You could then install a Wye & 1/8th bend to make an 1-1/2" vertical riser up to the lavatory . In this configuration the lav would be venting from the 3" horizontal and no additional vent would be required.