NOt all offset toilet drains are created equal...some are not all that bad to use, some are a constant clog waiting to happen. They should be avoided, if possible. If you use one, you want one that goes down at an angle, not a much shallower design that looks more like a spoon.
Code has some fairly specific rules about notches and holes through joists...plumbers tend to ignore them, but the structural guys may force the issue, if they are called in, too. Basically it is not okay to notch a joist...similar to an I-beam or a truss, it is the top and bottom that provide the strength - the middle (often hollow on trusses with a web) is just there to hold the top and bottom pieces in proper position for tension and compression for support.
It's hard for me to follow exactly what you want to do, but if I were doing it, I'd rather not use an offset flange and would either consider a different rough-in toilet (10 or 14") if I couldn't redo some of the framing to make a standard 12" one work. Keep in mind, if it isn't 12", your choice on toilets goes way down, and you may not be able to reuse your existing one, if that was in the plan.
A sketch may help. There is no issue with the flange being against a joist (i.e., there is no requirement for there to be a gap), but you do want to avoid notching things. A notch effectively makes the entire joist the equivalent of its remaining height. Whether that is still sufficient, you'd have to run the span tables, but most places don't overbuild much.