If I remember correctly, yes, that is the gun she killed herself with. And yes, in her second life, Carcel gave the gun to Oscar who then gave it as a gift to Ines.
In their previous life Carcel purposely gave it to the crown prince intending to have Ines use it! He told the crown prince some BS that would stroke his ego since crown prince was insecure that Ines was more talented at hunting than him.
I interpreted the last chapter as Ines being moved that Carcel is once again proving that he truly supports her, repeating the actions he had done in their past life. At that time he was one of the only people besides her older brother to be supportive of her hunting as it was a “man’s” hobby
Is that the very same gun she used to unalive herself in her previous life?
I remember it was given by the crown prince, but its not impossible since Oscar and Carcel are cousins