It's like "let's do something".

Давай пойдём - Let's go. Давай попробуем - Let's try. Давай не будем думать - Let us not think. Давай не пойдём - Let us not go. It's when you address one person (informal ты). Its structure is: Давай - singular imperative mood (the infinitive is давать), (мы) пойдём / попробуем - future perfective tense in the 1st person plural, будем думать - future imperfective tense in the 1st person plural. If you address two or more persons or use formal вы to one person, its structure is: Давайте - plural imperative mood. Давайте пойдём - Let's go. Давайте попробуем - Let's try. Давайте не будем думать - Let us not think. Давайте не пойдём - Let us not go.

Давай подумаем- один раз сейчас или в будущем/ соверш. вид глагола/. Давай думать- всегда, обычно регулярно. /несоверш. Вид глагола/ . Давай будем думать - обычно , регулярно , начиная с настоящего момента. Трудный вопрос, будет интересно прочитать ответы других.

Actually, there's only one way of saying and the only verb form - like in your first examples: "давай пойдем", "давай споем" etc. "Давай не думать" is just a short and more commonly used form from "давай не БУДЕМ/БУДЕШЬ думать" - "давай не бросать начатое" (давай не будем/будешь) бросать начатое" and so on

Your friend is right. "Давай не думать" has the meaning "вообще", "at all", "never". We use давай+infinitive when we want someone to stop doing this at all and it must be done at once and forever. This phrase also bears a slight meaning that it's natural to stop thinking about this and there this is the best way to do in the discussed situation. Hope it was helpful. I also want to give the links that will be interesting and helpful to you and other guys who learn Russian. http://www.learning-to-know.com/#!marathon/cx9x. http://www.learningtoknowrussian.com/ https://www.facebook.com/LearningToKnow