Is there any halachic problem with my dog coming into the sukka?
There is no clear halachic problem with this, except that you should make sure that the animal doesn’t relieve itself inside the sukkah. There are however Poskim who hold that it is preferable that it shouldn’t come into the sukkah, as it is an impure animal (b’heima temeiah), and therefore it isn’t respectful for it to be in the sukkah, similar to not bringing even clean pots into the Sukkah,
Shach Parshas Emor, Divrei Yatziv O:CH 2 274(3), Simchas Yisroel (Succah) 19-20, regarding a gentile entering the Succah.
R’ Y. Cahen shlit”a, Poskim.