Speed in a tube, how to calculate

You have a single pipe - let's say 1/2" - and you use this as an evaporator, submerged in water.
Tc = 40°C, Te= 0°C, let's say R134a
At the inlet a TEV, inlet SC liquid with 3K, outlet SH of 5K.
Second situation: SC of 20K.
I'm a little bit confused: in the 1st situation, we will have refrigerant with a lower quality than the situation with 20K. If we have 20K SC, and we used the outlet conditions to calculate the needed diameter of this coil (1/2"), will we then not have very high liquid speeds in the first parts of the tube, provoking a large DP in the first meters of the coil?

Yes you haven't. If you have the same outlet conditions, SC 20K and the same refrigerating capacity than you'll have less refrigerating mass-flow (less volume flow through TEV and after)... pressure drop in coil (1/2") will be less.