Wa alaykum salaam
Thank you for your question. Lying is one of those things that very few
people are any good at. Its eventual outcome especially for elaborate lies
is getting caught. The thing about getting caught is that not only have
you lost the trust of the people who you lied to, you are also more or
less forced at that time to explain the reasons for why you lied in the
Lying to your parents for the reasons given in the question is not allowed
jurisprudentially or morally in Islam. It is also not wise. Instead
perhaps be honest from the start and have a conversation with your parents
about it. The conversation may not result in the outcome that you
necessarily want, but try also through that effort to understand your
parents concerns. See there are some things in life that you can only
understand through experience. One of those things is parental love and
responsibility. It is not something that ends with the legal age and that
is why parents can sometimes seem overbearing especially before marriage.
Although we shouldn't let our lives be dictated by the perceptions of
others, those perceptions are at times a reality that faces us and we have
to pick and choose our risks. Those stigmas that your parents are worried
about are unfortunately there and your parred are only concerned with your
future happiness. My advise it to be patient and to know that even if you
don't always agree with your parents that you will never find people,
other than the Infallibles (as), that love you and are concerned with your
best interest more.
As for the dog try and keep it out of your room and pray there. Something
is only najis if you are sure it has become najis and in that case you can
make it clean in the ways explained further in the law books.
May Allah always protect you, grant you success in your education and
secure for you the most beautiful future.