Can angels appear in the shape of animals, such as a dog or cat?
An angel is pure spirit and, as such, has no shape. A body is a material thing; while material things change and die, pure spiritual beings do not. But for the purpose of interacting with us, an angel may at times assume a shape in order to fulfill the purpose God intends for the angel to accomplish.
If for the purpose of the angel’s visit to be accomplished the angel wants to be recognized, the angel would most likely assume a shape that would allow this. But the purpose of an angel’s visit may require an angel to go unrecognized. The author of Hebrews warns Christians, “Do not neglect to show hospitality to strangers, for thereby some have entertained angels unawares” (Heb 13:2).
God the Holy Spirit was willing to appear as a dove (cf. Mt 3:16, Mk 1:10, Lk 3:22, Jn 1:32). There is no reason in principle that an angel could not appear in animal form.