DC-DC buck converter low input voltage behavior
The inductance is way too high compared to suggested values. Refer to Fig.32 at p.17. A 22uH inductor is fine, but you placed 470uH. I know, the values suggested are given as &quot;minimum&quot;, but remember that increasing the inductance is not always the preferred way. A very large inductor may result in unexpected behavior.
I'll answer only the 1st question as the other two may be solved by decreasing the inductor to 22uH:

Note-6 at datasheet p.5: Voltage more than 4.65V is necessary for IC start. The IC can operate to 4.5V after IC start.