Thank you for the most interesting question. The solution rests in Regulation 650/2012, which even though it is EU-made, is not applied by Ireland, but it does apply in relation to Ireland.
The Regulation states that, for a person who was last habitually resident outside its geographical area of application, the EU member state (minus Ireland, UK and Denmark) where the person has immovable assets, i.e. a house, is competent for those assets (in terms of courts / notaries). Thus, the French notary public is competent for the house in France.
The French notary will recognize the Irish will as to formal requirements, because the Regulation provides that formalities for Wills can be those of the country where they were made, regardless whether in or outside the Regulation's application. The French notary will also apply Irish law to the estate, because the deceased had their last habitual residence in Ireland, unless Irish law sends back ("renvoi" through its conflict of law rules) to French law for the house. The exception to this exception is if the Will chooses expressly Irish law, in which case renvoi does not apply.
From the material point of view (i.e. the actual split), the Will has to be in conformity with the law to be applied (Irish or French, as the case may be). If it is so, then it will be applied in France as well.
To answer your second question, the notary will issue an Inheritance Certificate, on the basis of which you will be able to have the property registered in the names of the children.
Do let me know if you would like to provide further details for a more detailed response / if you have any other questions. Otherwise, I look forward to your rating, which is essential to my activity.
Dr I L Vlad