In most jurisdictions, the floor load is 40# PSF live load and 15#PSF dead load, but you could effectively reduce the dead load by 5# if you DO NOT attach the 2X8 to the 2X6, otherwise the assembly acts together and must meet the requirements. Typically, if I had to specify 2X8 in this instance, I would say use 2X8 SYP#2 @ 12" o/c, though I would prefer 2X10 SYP#2 @ 16" o/c.
If your local code allows the reduction of floor loads to 30# (many do) then 2X10 SYP#2 @ 24" o/c will just make it. In any instance, you must use 3/4" subflooring preferably glued to each member with either a high VOC PVA glue or something like Liquid Nail.
As far as attaching the new joists to the old...Don't ..let them act independently, Also, it might be a good idea to add a bearing block (A 2x4 laid flat on top of walls) under each new joist. This will prevent the flex from the floor from affecting the ceiling downstairs. Otherwise it could cause the ceiling to crack when pressure is applied to the ceiling gypsum or plaster.
And a little more to help you out... if the somewhat bad news is not enough.... For making your turn with the toilet drain, you should seek out a 90 degree street elbow and a closet flange to fit on the male end of the street elbow, You just will be able to make a legal turn with the toilet plumbing drain in 7 1/4" plus the floor decking. If you can't find that, then a regular elbow with a MALE closet flange will get you just about as close.
As far as doubling up on the floor decking, it should not be needed IF you use a standard tile underlayment board such as HardiBoard or WonderBoard. These come in 1/4" 3/8" and 1/2" thicknesses. If you are not using a ceramic, then you should be able to apply your sheet flooring directly over a properly prepared plywood floor.
Also, insulate between the floor and ceiling to soften the sound transfer from upstairs and to mask any pitiful singing that might emmanate from a happy camper using your new shower.
Good luck on your project and please DO NOT try to save money on the floor joists, in the long run, you will not be happy with the deflection and vibration when someone walks across the floor.