1. Davening B’yechidut in the Same Space

I am not sure what davening b’yechidut at school minyan looks like. Do you mean in the space where everyone is davening? What makes your davening b’yechidut and separate from the davening of the group?

If you are davening in the same space while others are conducting the service, you should not be answering devarim shebikdushah.

Other than that, the concern is messaging. Will people think that you are davening with them? What message will that send about Orthodoxy and halakhah’s position? Do they see you as an Orthodox Rabbi or as the School Rabbi? If the latter, then there might not be a messaging problem at all. It also might position you in a way to have more of an impact on the students. They may see you as more with them rather than standing apart.

It is really a judgment call of balancing the pros and cons here.

2. Accepting a Kibbud

This depends to which kibbud you are referring?

You can lead berakhot and Pesukei DeZimrah, and you can do hagbah and gelilah.

Regarding an aliyah: if there is a minyan of men present, then based on the principle of הכל עולין – hakol olin [i.e., “everyone may be counted among the seven” for Torah reading, including even a woman or minor, though the Gemara qualifies this because of kevod ha-tzibbur; see Megillah 23a], an argument could be made that the role of mechitzah is not as central here as we understand it to be in the rest of davening. Perhaps bi’shaat hadechak (in extenuating circumstances) and for chinukh purposes, one could allow receiving an aliyah in such a context.

However, beyond the halakhic concerns, there are serious policy considerations in getting an aliyah in a mixed space. I would avoid it if at all possible.

3. Responding to Kaddish, Barchu, Kedushah

This will depend on how many men are present. If there are fewer than 10 men, you should not respond to Kaddish, Kedushah, or other devarim shebikdushah. If, however, there are 10 men present, the following guidelines apply:

Kaddish Yatom requires 10 men, but it can be done in a mixed setting, on the model of Kaddish DeRabbanan at a siyyum.

For other ther devarim shebikdushah, including other Kaddishes, an argument could be made that they could be done with 10 men in a mixed setting and that the only real problem is Shemoneh Esrei and Chazarat HaShatz. However, I am not ready to make that claim.

Therefore, I would try to avoid all other devarim shebikdushah, even in the presence of 10 men, if it is a mixed space.

Does It Matter Whether a Man or a Woman Is Leading?

No. Regardless of whether a man or woman is leading the tefillah, you should not answer devarim shebikdushah in this setting.

Regarding Kedushah, if there are 10 men present, then yes, similar to a case where a woman made a siyyum at a home setting, and there were 10 men present, one could respond. However, in practice, given the broader concerns above, one should avoid responding to devarim shebikdushah in the mixed space.

4. Attending a Bat Mitzvah in a Conservative Shul on Shabbat

If you are only attending and not participating in any way, and you are not going to be using the microphone, then I think that is fine.