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With 0 ammonia & 0 nitrites, your tank is cycled. If ammonia/nitrite were present, they will show up on your tests even though your using Prime. Prime binds them- it doesnt remove them. Prime binds nitrates as well but you are still seeing them on your test. You should be able to stock your tank- just dont go overboard! I would monitor your water daily for the next week or two just to make sure your numbers stay stable.
You are fine to get fish as long as you do your wc to get your nitrates down (temp match & condition). You will just need to monitor your water for the next wk or two to make sure everything stays stable.
Prime will not give you false readings of zero. Prime binds ammonia (NH3) & converts it to ammonium (NH4). The API test detects any levels of NH3/NH4 present in your water. If your very concerned, you can wait to buy fish & continue to dose amm daily to make sure it steadily zeroes out.