I would recommend the following three:
- "We agreed on the price with the customer, but then he changed his mind."
- "I offered $1000 for the car and she agreed to that."
- "We agreed on $100 for the job."
In the first and third cases, the word "to" could be used instead of "on", but it would just sound more awkward, especially for the first sentence (and not as much for the third sentence).
In all three of these cases, using the word "at" as you suggested, would make the sentence sound a lot more awkward. The phrase "agree at a price" seems to imply to me that there was some back-and-forth bargaining that ended at some final value, and in that case the sentence would look like this (for example):
"After several hours of back-and-forth negotiation, we agreed at the final price of $100".