How would dirty fighting and outflank interact with each other if I didn't have improved grapple. For example, dirty fighting states:
When you attempt a combat maneuver check against a foe you are flanking, you can forgo the +2 bonus on your attack roll for flanking to instead have the combat maneuver not provoke an attack of opportunity. If you have a feat or ability that allows you to attempt the combat maneuver without provoking an attack of opportunity, you can instead increase the bonus on your attack roll for flanking to +4 for the combat maneuver check.
Could I use the ability from dirty fighting which allows me to not provoke an AOO and stack that with outflank also for a +4 to flanking while grappling? Or would it be +2 to grappling at that point?
No, if you have forgone your bonus to flanking, outflank no longer has anything to increase.
They don't work together very well at all, because even if you don't provoke (because of a feat or ability) and use dirty fighting to increase you flank bonus to +4 it overlaps with outflank as both set it to a fixed value rather than increasing it.
What if you also have Improved Grapple? Do the two +4s stack?
Outflank and Dirty Fighting do not stack because they both increase the same thing, flanking bonus, to a set amount of +4 (and wouldn't stack anyway even if worded differently due to Pathfinder's proscription against station bonuses from the same type, or flanking in this case).
Improved Grapple could then improve either attack value by +2 (for a net result is +6) since its bonus is untyped.