Am I right in thinking that this can be used as over discharge protection when connected to the Sunflower: Solar Power Manager 5V module?

Can this be used with the solar power manager?

How does the battery charge while using this?

Hello,
will this board work reliably with TPL5110 timer circuit (this board + MCU will be a without power for certain period of time)?

Thank you.

Yes, it is to protect charging process of the battery

No, it is only for Lipo battery.