If my neighbor’s tree leaned against a wall and cracked the block wall between us, what to do if he wants us to pay for half of the the $500 repair otherwise he wants us to take him to court?

Whether you have a case depends upon whether, prior to the damage being done, the neighbor had notice or knowledge that a problem was developing or that a significant risk existed. If he did, then he would most likely be negligent, or financially responsible, due his negligence, or carelessness, in not addressing the problem.

However, if he lacked knowledge of a risk or situation--say the tree, weakened by a recent storm, leaned up against the wall without prior notice; or this tree was growing so that it hurt the wall, but the fact that it was damaging the wall was not obvious or visible until the damage was done--then your neighbor would not be liable, since he did nothing wrong, and would not have to pay anything.

Also bear the following in mind: if the cost is $500 and your neighbor has offered to pay $250, you're only trying to get another $250 out of him. Court, even small claims court, costs something (usually around $30 for small claims) and also takes time--your time to file and try the case. Is it worthwhile to spend $30 and hours of your time to *maybe* get an extra $250--but also to maybe get nothing, if the court decides your neighbor was not at fault?

From what you have written I fail to see what liability your neighbor would have for a tree causing damage to your home under the laws of all states in this country. The damage is caused by an act of nature. I suggest that you make a claim to your homeowner's insurance carrier for the damages and take the $500 offered you by the neighbor.