Islam upholds the rights of animals to kind treatment and justice the same as it does for human being. Animals should not be abused or taken for granted. There are so many hadiths on this topic but let me just quote three of those to get my point across:
Narrated Abu Huraira:
Allah's Messenger (ﷺ) said, "While a man was walking on a road. he became very thirsty. Then he came across a well, got down into it, drank (of its water) and then came out. Meanwhile he saw a dog panting and licking mud because of excessive thirst. The man said to himself "This dog is suffering from the same state of thirst as I did." So he went down the well (again) and filled his shoe (with water) and held it in his mouth and watered the dog.
Allah thanked him for that deed and forgave him." The people asked, "O Allah's Messenger (ﷺ)! Is there a reward for us in serving the animals?" He said, "(Yes) There is a reward for serving any animate (living being) ."
حَدَّثَنَا إِسْمَاعِيلُ، حَدَّثَنِي مَالِكٌ، عَنْ سُمَىٍّ، مَوْلَى أَبِي بَكْرٍ عَنْ أَبِي صَالِحٍ السَّمَّانِ، عَنْ أَبِي هُرَيْرَةَ، أَنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم قَالَ " بَيْنَمَا رَجُلٌ يَمْشِي بِطَرِيقٍ اشْتَدَّ عَلَيْهِ الْعَطَشُ، فَوَجَدَ بِئْرًا فَنَزَلَ فِيهَا فَشَرِبَ ثُمَّ خَرَجَ، فَإِذَا كَلْبٌ يَلْهَثُ يَأْكُلُ الثَّرَى مِنَ الْعَطَشِ فَقَالَ الرَّجُلُ لَقَدْ بَلَغَ هَذَا الْكَلْبَ مِنَ الْعَطَشِ مِثْلُ الَّذِي كَانَ بَلَغَ بِي، فَنَزَلَ الْبِئْرَ فَمَلأَ خُفَّهُ، ثُمَّ أَمْسَكَهُ بِفِيهِ، فَسَقَى الْكَلْبَ، فَشَكَرَ اللَّهُ لَهُ فَغَفَرَ لَهُ ". قَالُوا يَا رَسُولَ اللَّهِ وَإِنَّ لَنَا فِي الْبَهَائِمِ أَجْرًا. فَقَالَ " فِي كُلِّ ذَاتِ كَبِدٍ رَطْبَةٍ أَجْرٌ ".
Reference: Sahih al-Bukhari 6009
In-book reference: Book 78, Hadith 40
USC-MSA web (English) reference: Vol. 8, Book 73, Hadith 38
https://sunnah.com/bukhari/78/40
In another hadith:
‘Abd Allaah bin Jafar said “The Apostle of Allah(ﷺ) seated me behind him(on his ride) one day, and told me secretly a thing asking me not to tell it to anyone. The place for easing dearer to the Apostle of Allah(ﷺ) was a mound or host of palm trees by which he could conceal himself. He entered the garden of a man from the Ansar(Helpers). All of a sudden when a Camel saw the Prophet (ﷺ) it wept tenderly producing yearning sound and it eyes flowed. The Prophet (ﷺ) came to it and wiped the temple of its head. So it kept silence. He then said “Who is the master of this Camel? Whose Camel is this? A young man from the Ansar came and said “This is mine, Apostle of Allah(ﷺ).”
He said “Don’t you fear Allah about this beast which Allah has given in your possession. It has complained to me that you keep it hungry and load it heavily which fatigues it.”
حَدَّثَنَا مُوسَى بْنُ إِسْمَاعِيلَ، حَدَّثَنَا مَهْدِيٌّ، حَدَّثَنَا ابْنُ أَبِي يَعْقُوبَ، عَنِ الْحَسَنِ بْنِ سَعْدٍ، مَوْلَى الْحَسَنِ بْنِ عَلِيٍّ عَنْ عَبْدِ اللَّهِ بْنِ جَعْفَرٍ، قَالَ : أَرْدَفَنِي رَسُولُ اللَّهِ صلى الله عليه وسلم خَلْفَهُ ذَاتَ يَوْمٍ فَأَسَرَّ إِلَىَّ حَدِيثًا لاَ أُحَدِّثُ بِهِ أَحَدًا مِنَ النَّاسِ، وَكَانَ أَحَبُّ مَا اسْتَتَرَ بِهِ رَسُولُ اللَّهِ صلى الله عليه وسلم لِحَاجَتِهِ هَدَفًا أَوْ حَائِشَ نَخْلٍ . قَالَ : فَدَخَلَ حَائِطًا لِرَجُلٍ مِنَ الأَنْصَارِ فَإِذَا جَمَلٌ فَلَمَّا رَأَى النَّبِيَّ صلى الله عليه وسلم حَنَّ وَذَرَفَتْ عَيْنَاهُ، فَأَتَاهُ النَّبِيُّ صلى الله عليه وسلم فَمَسَحَ ذِفْرَاهُ فَسَكَتَ، فَقَالَ : " مَنْ رَبُّ هَذَا الْجَمَلِ، لِمَنْ هَذَا الْجَمَلُ " . فَجَاءَ فَتًى مِنَ الأَنْصَارِ فَقَالَ : لِي يَا رَسُولَ اللَّهِ . فَقَالَ : " أَفَلاَ تَتَّقِي اللَّهَ فِي هَذِهِ الْبَهِيمَةِ الَّتِي مَلَّكَكَ اللَّهُ إِيَّاهَا، فَإِنَّهُ شَكَى إِلَىَّ أَنَّكَ تُجِيعُهُ وَتُدْئِبُهُ " .
Reference: Sunan Abi Dawud 2549
In-book reference: Book 15, Hadith 73
English translation: Book 14, Hadith 2543
https://sunnah.com/abudawud/15/73
Narrated Abdullah ibn Mas'ud:
We were with the Messenger of Allah (ﷺ) during a journey. He went to ease himself.
We saw a bird with her two young ones and we captured her young ones. The bird came and began to spread its wings. The Messenger of Allah (ﷺ) came and said: Who grieved this for its young ones? Return its young ones to it. He also saw an ant village that we had burnt. He asked: Who has burnt this? We replied: We. He said: It is not proper to punish with fire except the Lord of fire.
حَدَّثَنَا أَبُو صَالِحٍ، مَحْبُوبُ بْنُ مُوسَى أَخْبَرَنَا أَبُو إِسْحَاقَ الْفَزَارِيُّ، عَنْ أَبِي إِسْحَاقَ الشَّيْبَانِيِّ، عَنِ ابْنِ سَعْدٍ، - قَالَ غَيْرُ أَبِي صَالِحٍ عَنِ الْحَسَنِ بْنِ سَعْدٍ، - عَنْ عَبْدِ الرَّحْمَنِ بْنِ عَبْدِ اللَّهِ، عَنْ أَبِيهِ، قَالَ كُنَّا مَعَ رَسُولِ اللَّهِ صلى الله عليه وسلم فِي سَفَرٍ فَانْطَلَقَ لِحَاجَتِهِ فَرَأَيْنَا حُمَّرَةً مَعَهَا فَرْخَانِ فَأَخَذْنَا فَرْخَيْهَا فَجَاءَتِ الْحُمَّرَةُ فَجَعَلَتْ تَفْرُشُ فَجَاءَ النَّبِيُّ صلى الله عليه وسلم فَقَالَ " مَنْ فَجَعَ هَذِهِ بِوَلَدِهَا رُدُّوا وَلَدَهَا إِلَيْهَا " . وَرَأَى قَرْيَةَ نَمْلٍ قَدْ حَرَّقْنَاهَا فَقَالَ " مَنْ حَرَّقَ هَذِهِ " . قُلْنَا نَحْنُ . قَالَ " إِنَّهُ لاَ يَنْبَغِي أَنْ يُعَذِّبَ بِالنَّارِ إِلاَّ رَبُّ النَّارِ " .
Grade: Sahih (Al-Albani)
Reference: Sunan Abi Dawud 2675
In-book reference: Book 15, Hadith 199
English translation: Book 14, Hadith 2669
https://sunnah.com/abudawud/15/199
Now I noticed that you specifically mentioned cats and dogs in your post.
Having cats as pet is not forbidden in Islam at all. In fact, one of the companion loved cats so much that he was nick names as father of cats [ Abu Huraira - may Allah be pleased with him ]. There is also a hadith of a woman who entered hell fire because of her bad treatment of her per cat.
Narrated Ibn `Umar:
The Prophet (ﷺ) said, "A woman entered the (Hell) Fire because of a cat which she had tied, neither giving it food nor setting it free to eat from the vermin of the earth. "
حَدَّثَنَا نَصْرُ بْنُ عَلِيٍّ، أَخْبَرَنَا عَبْدُ الأَعْلَى، حَدَّثَنَا عُبَيْدُ اللَّهِ بْنُ عُمَرَ، عَنْ نَافِعٍ، عَنِ ابْنِ عُمَرَ ـ رضى الله عنهما ـ عَنِ النَّبِيِّ صلى الله عليه وسلم قَالَ " دَخَلَتِ امْرَأَةٌ النَّارَ فِي هِرَّةٍ رَبَطَتْهَا، فَلَمْ تُطْعِمْهَا، وَلَمْ تَدَعْهَا تَأْكُلُ مِنْ خِشَاشِ الأَرْضِ ". قَالَ وَحَدَّثَنَا عُبَيْدُ اللَّهِ عَنْ سَعِيدٍ الْمَقْبُرِيِّ عَنْ أَبِي هُرَيْرَةَ عَنِ النَّبِيِّ صلى الله عليه وسلم مِثْلَهُ.
Reference: Sahih al-Bukhari 3318
In-book reference: Book 59, Hadith 124
USC-MSA web (English) reference: Vol. 4, Book 54, Hadith 535
https://sunnah.com/bukhari/59/124
Regarding dogs, Islam does not allow to have dogs inside a house but one can have dog in their backyard. If you think about it, this is also related to animal rights. Dogs by nature need an open space to live and by putting them inside a house as per, we humans are actually violating animal rights. Think about how did you feel when you are forced to be quarantined and many people are upset because they can't socialize as they used to do.
Anyways I hope that answers your question. If there is anything still confusing, please feel free to ask.
All the best!
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