Is it possible for a commercial pilot to set a minimal rate of climb - for example, 350' NM climb rate, target heading of 200 degrees and target altitude of 2000 feet above MSL - before getting radar vectors from ATC when they hit the target altitude?
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.