How to demolish old stone walls safely?
Taking down the wall safely is not the only issue. You wrote in comments:

there's no legal problem here because we own all the land surrounding the house that isn't ours, and the connection between the house and the old wall is actually on our property and not technically legal.

My perspective is that of a retired US lawyer, wholly unfamiliar with French real property law. In many legal systems, however, an activity on your own land (such as demolishing the wall) that affects or impairs your neighbors' land or structures thereon could create a serious legal problem for you. In US law, the underlying (pun!) concept is called subjacent support. Changing the land's drainage pattern could similarly affect others' property, and create similar liability.
Before you take any action on the ground, I would consult someone knowledgeable about these sorts of matters in your locality. An attorney or local notaire, or at least someone who has personal experience in real property construction and demolition. The person should not be someone who is involved in proposing, or doing, or contracting the work you want - that person has a built-in conflict of interest.
Removing the walls may be entirely doable. But doing so also may create serious and persistent problems, physical, fiscal, or both. You will be better off knowing before you act.