Is it true that good accuracy of data ALWAYS indicates good precision?

Answer 1

No. Consider.
Actual value = 20.00%

Values reported:
20.00%
30.00%
10.00%
Avg = (20.00+30.00+10.00)/3 = 20.00
Great accuracy.
Lousy precision. In fact with precision like this no one would even report the values; most likely the work will be repeated.