Hi Mark! don't make me is 别让我bié ràng wǒ. 别让我 +do something. 别让我can be also translated as don't let me. You can say 别让我走，bié ràng wǒ zǒu, don't let me go. 别让我伤心bié ràng wǒ shāng xīn， don't make me sad. 伤心here is a verb, to be sad, not an adjective.Please be noted that it is only followed by verb or verb clause. For example, you can't say 别让我漂亮piào liàng，漂亮is an adjective here. Hope this helps!

There is no such phrase, or phrase pattern, in Chinese as I'm afraid of, they allverify from one to another. As to the 2 examples you have asked, the first one is 别气我 , where 气 is the verb form of 'make someone angry'. The most native way of saying the second phrase is, 别让我丢人现眼, but as you can see, these 2 phrases are quite different in respect of grammatical structure

Don't make me...= 別讓我...