“Traduttore, traditore”

Quick answer:
The not-so-bad translation, here, is indeed “appropriate”.

Expanded answer:
The last definition of “propre” in your dictionary is the one that is used in the text. To get the equivalent meaning of “propre”, you would have to wrap “appropriate” and “specific quality/attribute” together in one word.

Appropriate doesn’t seem to me perfectly appropriate here…

The first three instances of propre are adjectives that are close to your dictionary definition: Qui rend exactement l’idée

I would translate them with "able to" or "suitable", e.g. :

… deux sortes de problèmes fondamentaux, c’est-à-dire propres à faire comprendre de quelle manière se comporte un corps quand il se déforme…

… two kinds of fundamental problems, i.e. able to make clear how a body behave upon deformation…

Il n’en est vraisemblablement pas qui soient plus propres à nous faire connaître de quelle manière…

Probably none are more suitable to tell us which way…

… les rend également propres à exprimer les conditions dites aux limites…

… make them also suitable to express the so-called "to limits" conditions…

The last occurence match the first definition, "own" or "proper".

… le principal des termes qui expriment l’action propre de ces modes est en raison inverse du cube de la distance.

… among the terms that express these modes proper action, the main one is in inverse ratio to the cube of the distance.

It seems you had the best answer and the best obersvations. In your three context, you seem to have found the proper way of translating propre… Your dictionary does not seem to have properly targeted this meaning but it comes close as previous answers indicate. For example, convenable is not really appropriate for qualités propres au commandement, one would expect more than convenable.

At any rate, you could look at the definition of proper in English which should be close enough…