Learn Russian: Можно мне трогать этот камень?

The word можно is an impersonal word that roughly means 'it is allowed' or 'it is possible'. In Russian, instead of saying 'I may', you literally express 'to me it is allowed'. That is why we use the Dative case pronoun мне instead of the Nominative я.

Does the word order matter here? Can I say Мне можно... instead?

Yes, you can say Мне можно трогать этот камень? Russian word order is flexible. Starting the sentence with Можно places a bit more emphasis on the core question of permission ('Is it allowed?'), which is very natural for this type of question.

Why use the imperfective verb трогать instead of a perfective verb?

When asking for general permission or establishing rules, Russian prefers the imperfective aspect. Using можно with the imperfective трогать focuses on the action itself rather than the completion of a single event, which fits perfectly when asking if an action is permitted at all.

Why doesn't the ending of этот камень change, since it is the object being touched?

Because it is the direct object of the verb трогать, the phrase is in the Accusative case. However, камень is a masculine inanimate noun (a non-living thing), meaning its Accusative form looks exactly like its Nominative dictionary form. The same rule keeps the modifier этот looking exactly like its dictionary form.