I will be participating on a summer program this summer with egalitarian davening, including women leading, laining, having aliyot, etc. Can I respond amen to their brachot? Can I respond to their kedushah? To their aliyot? If I was a gabbay, could I call up a girl? I cannot think of other questions that might come up, but if anyone can think of other issues please add them.
Another thing to consider is dealing with the group's definition of what counts as a minyan/quorum. For example, a nice (and real) question is what would you do if Halacha/your psak says that there is nowhere close to a minyan but at some important point the group holds that there is exactly a minyan with your presence.
Even without having a role, you can't always be a neutral observer.
Rav Moshe said to distance oneself from such minyanim (OC 4 91:6).
Rav Henkin permitted davening in a shul without a mechitza under limited circumstances. He writes:
"Every individual should live in a place of observant Jews if possible. However, if this is not possible, we should not be strict concerning these matters because it will lead to a potential catastrophe.
However, if the place itself is corrupt in that it has mixed-seating, it has already been established that it is preferable to pray by yourself at home. But, if this is the only synagogue in the area and you will always have to pray at home, you must examine the situation and evaluate the corruption versus the hope that through the involvement of the observant in this congregation, the community will become Orthodox. Yet, in all situations you must reprimand them if you pray in their midst."
I'm not sure how this ruling would apply to the summer program you are participating in, since R. Henkin was talking about only mixed seating, but not necessarily mixed participation, but some of the factors to consider are mentioned in his response.