If someone mistreats their animals or pets, is that punishable?
It is impermissible to treat Animals bad, and the punishment is in the hereafter from Allah, but there is no haad or punishment in life, Also there is a reward from Allah for being kind to animals, Wallah A'laam.
Narrated 'Abdullah bin `Umar:
Allah's Apostle said, "A woman was tortured and was put in Hell because of a cat which she had kept locked till it died of hunger." Allah's Apostle further said, (Allah knows better) Allah said (to the woman), 'You neither fed it nor watered when you locked it up, nor did you set it free to eat the insects of the earth."
حَدَّثَنَا إِسْمَاعِيلُ، قَالَ حَدَّثَنِي مَالِكٌ، عَنْ نَافِعٍ، عَنْ عَبْدِ اللَّهِ بْنِ عُمَرَ ـ رضى الله عنهما ـ أَنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم قَالَ " عُذِّبَتِ امْرَأَةٌ فِي هِرَّةٍ حَبَسَتْهَا، حَتَّى مَاتَتْ جُوعًا، فَدَخَلَتْ فِيهَا النَّارَ ـ قَالَ فَقَالَ وَاللَّهُ أَعْلَمُ ـ لاَ أَنْتِ أَطْعَمْتِهَا وَلاَ سَقَيْتِهَا حِينَ حَبَسْتِيهَا، وَلاَ أَنْتِ أَرْسَلْتِيهَا فَأَكَلَتْ مِنْ خَشَاشِ الأَرْضِ ".
The Prophet(sallallahu alaiyhi wassallam),was asked if acts of charity even to the animals were rewarded by the Almighty. He replied: "Yes, there is a reward for acts of charity to every beast alive." (Narrated by Abu Huraira, Bukhari)
Mishkat Al-Masabih concluded from "Bukhari" and "Muslim" to the effect that: "A good deed done to a beast is as good as doing good to a human being; while an act of cruelty to a beast is as bad as an act of cruelty to human beings," and that: "Kindness to animals was promised by rewards in Life Hereafter" (Mishkat al-Masabih; Book 6; Chapter 7, 8:178).
In addition to the great answer by brother مجاهد
Abu Hurairah (May Allah bepleased with him) reported: Messenger of Allah (sallallaahu alayhi wa sallam) said, “On the Resurrection Day, the rights will be paid to those to whom they are due so much so that a hornless sheep will be retaliated for by punishing the horned sheep which broke its horns”. [Muslim].
So, if sheep will be judged for harming other sheep, how about humans?