All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
First of all, all perfect praise be to Allaah who guided you to Islam which is the greatest bounties of all, Allaah Says (which means): {Say, "Indeed, my Lord has guided me to a straight path — a correct religion — the way of Ibraaheem (Abraham), inclining toward truth. And he was not among those who associated others with Allaah."}[Quran 6:161].
Islam means submission to Allaah by abiding by His Commandments that were revealed, i.e. the Book of Allaah and the Sunnah [the actions approved, done or prohibited by] of the Prophet . Allaah Says (which means): {The only statement of the [true] believers when they are called to Allaah and His Messenger to judge between them is that they say, 'We hear and we obey.' And those are the successful.}[Quran 24:51].
It is reported in the Sunnah that it is forbidden to acquire dogs unless there is a need for it, like a hunting dog, a shepherd dog, or a farm dog. We have already issued Fataawa [plural of Fatwa] 86844 and 81421 on this. Indeed modern science has discovered some wisdom why it is forbidden in Islamic religion to keep a dog because the dog has germs that transmit dangerous diseases. So it is forbidden in Islamic Law to keep a dog unless there is a greater benefit than the harm of keeping it. It is enough an evil that a Muslim who keeps a dog, one Qirat (a given portion) of the reward of his good deeds is deducted daily, as clarified in the above Fataawa.
As regards wearing Hijab, it is an obligation on Muslim women as clarified in Fatwa 88490, so you have to abide by this obligation even if some people are not pleased. They will not benefit you when you will stand before Allaah for judgement. It might be that one day Allaah will make the people be pleased by you. The Prophet said: "Whoever seeks Allaah's pleasure by angering people, Allaah will suffice him from the people, and whoever seeks people's contentment by angering Allaah, Allaah will forsake him and leave his matters for the people." [At-Tirmithi] You have to know that whoever leaves something for the sake of Allaah, Allaah will replace him with something better. Allaah Says (which means): {And whoever fears Allaah — He will make for him a way out. And will provide for him from where he does not expect.}[Quran 65:2-3].
Finally, we advise you to keep in touch with the Muslim sisters in Islamic centres and seek their advice in relation to any difficulties you might encounter. Nonetheless, keeping a dog, even when it is forbidden and not covering the hair does not take the person out of the fold of Islam.
Allaah Knows best.