I didn't think of sending it back. But the tone & demeanor of the adjuster I dealt with, whoa he was very cocky. I don't want to deal with them at all. In fact, even before I filed a claim, when I called their 1-800 number to find out how to file, the agent who answered had major attitude-- hostile, his entire tone and demeanor. So I'm not expecting them to act in good faith at all. From the get-go they seem to see it contentious, and therefore preemptively take an adversarial tone and stance.Exhibits get presented at trial along with your testimony.
That's not evidence of anything the driver did. It's only evidence of what the street looks like when nothing is happening. Present it if you like, but it isn't likely to help you.
The position of the damage on the vehicles might give a hint to how the accident happened.
Send the check back with a letter declining the offer and attach a copy of your completed (but unfiled complaint) along with your estimate. Then see what the response is.
Just holding on to the check without cashing it can be construed as accepting the money. That you might not want to send the check back for fear of not getting it again implies that you have accepted it whether you cash it or not.