Is there a way to check if a script failed using another script, and if so, what the errors were?

It's possible using ScriptContext.Error, which passes the error message, traceback, and the script that errored

game:GetService("ScriptContext").Error:Connect(function(message, trace, erroredScript)
	print(erroredScript.Name .. " has errored!")
	print("Error message: " .. message)
	print("Trace: " .. trace)
end)

Note that will only fire when a script errors. If you want an event that fires with warnings or prints, you could also use LogService.MessageOut (however it doesn't send which script sent the message)