I bought an Haier w718 online from China. It is unlocked but they didn't put Google Play on it. If I root it or flash it will that relock the phone?
No, since it is factory unlocked, nothing will ever relock your phone.
Back in the days of the ZTE Blade there was a ROM that [sim] unlocked the handset, flashing another rom would remove that unlock, but like @Shywim says if it is factory unlocked it is unlikely that flashing another ROM will lock it and like @Izzy says it is also unlikely to happen in a custom rom