Commercial aircraft minimum and maximum climb angle and rate?
You are asking several different questions, but I can answer the first part easily enough.  

First, let's distinguish between climb rate and climb gradient:  climb rate is expressed as feet per minute, climb gradient is expressed in feet per nautical mile.  

The autopilot I am familiar with can be programmed for a target climb or descent rate, heading and altitude.  With ground speed and some math a FPM rate could be set that will achieve a desired gradient, but I am not aware of any systems where this can be actually set.

737 takeoff and climb performance will vary greatly depending on weight and other variables.