Do pets enter Paradise with their owners on the Day of Judgment, and has anything regarding this been mentioned in the Sunnah?

There is no evidence for animals entering Paradise. Rather, the evidence indicates that Allah will turn animals into dust on the Day of Resurrection, as stated in the long Hadith (definition) of the Trumpet: "For on that Day, the hornless will be given retribution from the horned, until there remains no grievance from one against another. Then Allah the Almighty will say: 'Be dust!'" It is also narrated on the authority of Abdullah ibn Amr and Abu Hurairah, as a mawquf narration: "When the retribution among the animals is finished, He will say to them: 'Be dust!'" and "He will say to the beasts and birds: 'Be dust!'"