No matter how strange to me some American Orthodox customs are, you can't have been given Holy Chrism, just a blessed oil. Yes, you can anoint each other at home, I wouldn't anoint dogs.
There are (at least) four kinds of oil in the Church.
1. Holy Chrism, which is consecrated infrequently by an autocephalous Synod in an elaborate ritual, including mixing in many different spices and essential oils. Holy Chrism is only used by the priest to administer the Mystery of Holy Chrismation, or by Bishops to consecrate a church or an antimension. It's never handed out, and no layperson should use it to anoint anyone or any thing.
2. Holy Oil, which is consecrated/blessed by a priest during the Mystery of Holy Unction for the healing of soul and body and the forgiveness of sins. There are some churches, especially in the diaspora, that will allow people to take some of this home with them, so they can anoint missing family members. However, some priests and bishops object to this.
3. Holy Myrrh, sometimes also called Holy Oil, which streams forth from relics or an Icon. Many churches throughout the Orthodox world will allow the faithful to bring this home to anoint their families.
4. Holy Oil, sometimes just called Oil, taken from a lampada
that is next to or in front of relics or a special Icon. Many churches throughout the Orthodox world will allow the faithful to bring this home to anoint their families.
It is most certainly inappropriate to anoint an animal with No. 1 or 2. I've never heard of anyone doing it with No. 3 or 4. Personally, I wouldn't, but there are traditions in which the priest will sprinkle Holy Water on animals.