Nice picture, but identification of the problem somewhat difficult!

Contact your buildings insurance people - they should send out a structural surveyor to ascertain whether the damp you have is caused, or connected with, the trees on your neighbour's property. If it is, your neighbour is liable for the costs of repair.

Yep, like Bamboo says, your neighbour is liable.

Just like branches, roots that grow from your neighbours land onto yours are trespassing. You can chop the roots along the boundary line and you do not need your neighbours permission to do so.

Roots can often lead to substantial damage e.g. by growing under the foundations of the house and causing them to become unsafe or by causing the soil to dry out resulting in subsidence. In these cases damage will be :-

1. The cost of repairs
2. The reduction in the value of the house if there is a loss on sale.
3. Any other expenses directly arising from the trespass.
4. In all likelihood the legal costs of the person who has suffered the damage.
5. An order by the court that the owner of the tree cuts back the branches or the roots or takes down the tree itself.

Subsidence/damage caused by tree roots will involve a claim, which will generally be a nuisance rather than trespass if it has to go to Court. Naturally a negotiated agreement with neighbours would be far preferable.

I hope this helps? The best course of action would of course be to have a polite word with your neighbour about the problem first. But being aware of your legal rights before hand is very helpful in case they're not too helpful ;o)