A: It would be very difficult to prove that a beneficiary designation was changed due to mental capacity or undue influence. Basically, whichever beneficiary is on the policy is unlikely to be changed after death. A lawyer would need to investigate the facts and case law for a more specific answer.
A: There is no clear answer to this question in New Hampshire, surprisingly. The Vermont Supreme Court case of Alvarez v. Katz has some guidance (Vermont law): https://law.justia.com/cases/vermont/supreme-court/2015/2014-385.html.
Practically speaking, if you cut without permission, you are risk the possibility of a claim against you (particularly if it would damage the health of the tree). Best to reach some sort of written agreement/permission with your neighbor.