FWIW a general rule is the Brits have lousy teeth, your Dog could be quarantined for months.. You also may have a quarantine from country to country also..
Unless you know your going to stay leave him in the states with parents or some one as a loaner.. It will be real expensive to transport him to europe also airline charge an arm and a leg..
Lots of countries will not give you a Work visa unless you have a job before you come over.. Something you would want to contact those countries you plan on visit ask about visas and your dog there are consulates throughout the US and I'd think many would have them in SEA..
You may want to look at Canada the eastern part if your looking for speaking french..
western part if your looking for America north..
the cost of living in Europe it also on the moon, so long term you could be in trouble.
Not so sure that hostels will take dogs either. something you may want to check out..
Australia or New Zealand could be a new frontier for you, kind of the best of old Europe and America.