Location of Shower P Trap

Hi -
This is my first post. I'm a DIY'er trying to put a new shower in the upstairs bathroom of my daughter's older home (it had a lead trap and lead drain line) - I think that makes it Oooold.
Anyway - I'm going to replace all that with plastic. My question: space under the location of the new shower is very tight. Can I connect to the shower drain with an elbow, then place the trap about 3 feet to the side? I would then attach the vent before going straight down through the downstairs wall to the basement.
Question #2 (as long as I'm at it): is there a product that enables me to connect this plastic drain line to the existing cast iron pipe in the basement? Maybe a rubber "boot" or something?

Thanx

Placing a trap that far away from the opening of the fixture will result in horrible smells emitting from the drain over time. Codes dictate the maximum distance a trap shall be located from the fixture it serves. Same goes for the venting and line size. Sometimes copying the install that the lead was done in is wrong, therefore it is recommended that you take pictures and show us what exactly your situation is. It might be a great deal easier with numerous opinions off of this site.