At this stage we have not been advised that people may travel with deferred adjudication status. I do not recommend it.
Are you referring to the regular Deferred Action or DACA (Deferred Action for Childhood Arrivals)?
[This answer is for general purposes only; it does not constitute advice and does not establish an attorney-client relationship.]
Only if you are issued an advance parole BEFORE you travel.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.
Probably yes. Once DACA is approved, you can file an I-131 application for advance parole. You must justify it for educational, work, or humanitarian reasons. I think that a church mission could fit into those somehow.