A similar idiom in meaning would be "better to use the carrot than the stick" This refers to the concept of a rabbit doing something you want for a reward (carrot) versus a rabbit doing something you want out of fear of being hit (stick).

Yes, you can! The phrase is familiar enough that we can adapt it. Another well known example of a shortened proverb is "great minds think alike". Originally, the full sentence adds "...and fools seldom differ", but not many people are aware of that extra phrase.