It’s usual for the phone in normal charging operation to boot up until the Fairphone logo appears and then switch to the blue battery gauge which should display the “charge” level of the battery. This is the least default boot level just for charging.
Yes, there are generic battery chargers for different phone batteries.
In normal operation without battery but connected to a charger it should act in a loop permanently restarting the phone after showing the Fairphone logo.
If not done so yet, feel free to read more about battery issues here.
You could look out for one of our fairphoneangels in your area to swap batteries/phones for testing.
This could be related to a too loose battery compartment not properly contacting the battery. Please read more about this in the above mentioned batteryguide.
Feel free to use the search function here as well. There is plenty of valuable information to find about the FP2.