I'm currently using Daylio. It works fine, but, given the sensitive nature of the information, I want something more private/trustworthy. The following is what I am looking for in the app:
Essentials:
Android app.
Support custom tags and notes when recording a mood.
The ability to add a mood for a specific day/time other than current (for example, if I miss an entry).
Support multiple entries per day.
Nice-to-haves:
Visualizations of the moods, and other data, over time
Not sure if it has every feature you're looking for, but I was able to transition from Daylio to Daily You and be ok with it. Can't track what activities you did with buttons, but can track your mood every day with a journal entry for each day, and an image if you want. There's even a Python script to convert your entries from Daylio into Daily You!