You don't need to say both. You can just say: We plan on staying there for three nights. or We plan on staying there for four days. However, it's not wrong to say: We plan on staying there for four days and three nights OR We plan on staying there for three days and four nights. Both are okay.

I've never heard that expression before, but I would assume it means that person A is asking person B if they would be willing to commit a dime to buy something.

Question 1: They have the same meaning and both are grammatically correct. Question 2: They have the same meaning and both are grammatically correct. Just two slightly different ways of saying the same thing :)

People with a southern accent will pronounce "thing" as "thang."

Like Paul said, "set up" usually means setting two friends up on a date. Introduce just means you want them to meet each other, as friends.

"I am attending a course next next semester" sounds odd to me. "I am taking a course next semester" would sound better. Its a subtle difference, but I have never heard someone say that they will "attend a course." It might not be wrong gramatically, but it just sounds odd. The word "attend" is usually associated with a specific event. Taking a course is more accurate because it's a regular commitment that you have adopted. However, to answer your question, we can say "I am taking a course next semester" because it implies that you have already committed to it. In essence, it says that you are, at the present moment, enrolled in a course. You may not have gone to a class yet, but you are already enrolled in it. Thats why we can use the present tense in this situation.