Question

2) The temperature of the mixture of ice and water remains
till the ice melts completely​

Answer 1

Yes , due to latent heat of fusion .

Answer 2

Pure water and ice, kept insulated from the warm outside world, come to equilibrium over time. On a molecular level, water molecules are freezing onto the ice at the same rate as they are melting off of it. The entire water/ice solution is at the melting/freezing point, 32°F (0°C).