OP,
The goal is to cool the syngas from 300°C to 60°C, with the syngas on the tube side and cooling water on the shell side. Why? Have you considered the pressure of 300degC water? What are you doing with this heat?
How many shell and tube heat exchangers would be required for this task? 1, 1000, 10,000..... You have not given the process variables needed to size nor design an exchanger.
Would one be enough, or would I need two or three? See previous
Additionally, how should I set the temperature range for each exchanger? This is only one part of the process variables, many others will need to be determined for design.
What are you doing with this heat? If you truly are just rejecting it, why use water and not ambient air? Why are you not reusing this heat?
For HX (heat exchanger) design, you will need the properties of your syngas along with the coefficients to determine transient properties at different temperatures and pressures, likewise for water. Then you will need to do a mass and energy balance to determine the mass flux entering and leaving the system and heat flux, entering and leaving the system. From there you will need to determine the material properties and geometries of the HX,,, ie... wetted surface area, differential pressures, flow rates, expansion coefficients....
The following link describes only the process data needed for HX sizing prior to the mechanical requirements being determined.
I suggest you go back to the very beginning of your design and perform a mass and energy balance around the entire system (a syngas plant I assume), then work your way through the system defining process conditions. Only then can you get to the point where one can consider HX design.