if you have both liquid and solid water at equilibrium then you have zero degrees Celsius. Pressure has minimal effects …at plus or minus 0.5 atmosphere. of course if you go to a hundred or a thousand atmosphere then there is an effect of pressure.
Small pieces of ice will equilibrate their temperature faster in water.
Surface tension has minimal effect on melting temperature unless you go to extremely small pieces of ice meaning less than one micron, …which is not possible to achieve anyway because such small ice pellet with fuse rapidly to form larger ones.