Do these products allow charging and powering a device at the same time?
https://core-electronics.com.au/solar-lipo-charger-3-7v.html
https://core-electronics.com.au/usb-lipoly-charger-single-cell.html
https://core-electronics.com.au/adafruit-micro-lipo-w-microusb-jack-usb-liion-lipoly-charger-v1.html
https://core-electronics.com.au/usb-liion-lipoly-charger-v1-2.html
Also I can just connect a 5V wall charger to
https://core-electronics.com.au/solar-lipo-charger-3-7v.html
Instead of a solar panel and it should be fine right?
Yes
Yes
Yes
Yes
This device is intended to be used with a 5V power supply and not a solar panel.
Hopefully all that helps!