Is Dog allowed in Hindu household and Hinduism?

No! Dogs are not supposed to be kept inside the house. This is true for cocks and hens also. When dogs or cocks are in sight, whatever offerings you make to the Gods, Gods wouldn’t accept them. Rearing wild animals like cats is a sin too. 

All offerings done to Gods, in the sight of dog are consumed and taken away by ‘krodhavasha-s’ who are a class of rAkshasa-s’. There is no place in heaven for them who raise dogs because all the merits of theirs get taken away by the krodhavasha-s, hence such people land in hell.

Mahabharata says the following in this regard:

“Dhaumya said, 'Broken utensils, broken bedsteads, cocks and, dogs, as also such trees as have grown within the dwelling houses, are all inauspicious objects. In a broken utensil is Kali himself, while in a broken bedstead is loss of wealth. When a cock or a dog is in sight, the deities do not eat the offerings made to them”. (MBH Anushasana parva Section CXXVII)

Indra also says the same to Yudhishthira as follows:

“Indra said, ‘There is no place in Heaven for persons with dogs. Besides, the (deities called) Krodhavasas take away all the merits of such persons. Reflecting on this, act, O king Yudhishthira the just. Do thou abandon this dog. There is no cruelty in this.’”. (MBH Mahaprasthanika Parva Section 3)

Dogs are kept only by Sudras.

Touching a dog requires one to perform achamana to purify oneself. Vedas should not be repeated when a dog is barking.

The Bhagavata Purana says that the twice-born men who keep dogs in the home will go to a hell where they are made to run and shot with arrows.

In Hinduism dog is the most inauspicious of animals, to be kept away from wedding altars and holy sites. Vedas must not be recited if a dog's barking is heard. Dog's howling is a bad omen aswell.

If you follow Pitra rituals (as most Temples 🚩) then dogs are strictly not allowed.

Dogs are considered as harbingers of bad luck. This is because we get attached to dogs and in Hindu philosophy “Attachment entraps”. For ex. In the Bhagavata Purana is the story of Bharata who is a hermit in the forest. He gives up everything but slowly gets attached to a deer. As a result, he is unable to attain moksha. He is reborn as a deer, trapped once more in the cycle of rebirth. Attachment entraps: this is a key maxim in Hindu philosophy.

Now visualize a dog looking at you with eagerness and affection – it adores you and its behavior melts your heart. If you have a pet dog, you will know that the dog constantly seeks validation from you. Give it that attention it craves and it will wag its tail, don’t give it and it will whine.

Now visualize a hermit surrounded by dogs. Does he surrender to the affection of the dog? Does he, like Bharata getting attached to a deer, get attached to these dogs? If he seeks to break free from the cycle of rebirths, he must transcend the urge to get attached. The dog is the ultimate temptation, because the dog gives its master absolute unconditional love and devotion. Nothing is more tempting, not even the dance of damsels known as Apsaras. When Dattatreya, the mendicant, walks with four dogs around him – it indicates his perfect detachment. The dogs follow him but he does not lead them

One of the enemies of human being according to Shrimad Bhagawad Geeta is Moh (attachment) along with Kaam (desires), Krodh (anger), Lobh (greed), Mad (pride or ego) and Matsar (jealousy).

Only allowable pets are the holy Cows in the home.

Animals are born to workout their Karma, non interference in their life is good for us. Unless they become dangerous to our life, that needs action, leave them to their natural life and environment. Rearing, selling, buying, killing, cooking, eating animal flesh are considered sinful for human. It is permitted only under life and death situation, extreme starvation. Help animals in times of distress but do not interfere with their nature of existence.

That being said, Dog is vehicle of Kal Bhairav, a Rudra dangerous form of Mahadev. Also, Lord Dattatraya who was an incarnation of the Trimurti (Brahma, Vishnu and Maheswar) is always followed by four dogs which symbolically represent the four Vedas. Lord Yama had taken the form of a Dog to accompany Yudhistira during their final journey. He was testing the dharma of Yudhishthir. 

Indra Dev (the king of Devas) has a dog named Sarama. This dog is supposed to have helped the Devas bring back their stolen cows by demons. Sarama is the mother of all dogs and is the daughter of Daksha (sister of Sati). But her children Sarameya are the vehicles of Yama and Yama doots. That's why dog's are said to "see the dead".