We've taken dogs into Mexico ... no further experience. Mexican govt. never asked about our Dog. Re-entering USA US customs asked if we had vaccination certificates ... we said YES, they never asked to see them.
(We had all Vet docs with us of course: Rabies, distemper, et al)
Every country has their own rules regards pets ... probably can be done in most cases.
"Lew and Punky" traveled S. America on Lew's KLR & Vstrom. His little Yorky went everywhere.
More Med skills you have the better!
Language is important too. If you don't speak Spanish, you won't be much help & may not be able to get paid. Learn it. (you'll be the first Aussie!

)
Prepare to be shocked at lack of "sterile technique" outside western countries. Also, the lack of basic equipment and common medications. Public clinics can be especially hard hit, Due to corruption, supplies get "sidetracked" to private clinics for those who can pay. It's a mess ... but that does not mean you shouldn't TRY to help out! They NEED YOU!
Not sure how you can get paid or best places to get work. Outside of cities there is tremendous need for medics, nurses and Doctors. You could try to hook up with NGO's. I worked for two (film crew) going round documenting work done in various countries. If an NGO has a clinic, it's likely they'll have translators and foreign doctors working there. The Docs mostly come and go, so lots of turnover. That might be your best bet ... as NGO's have some money.
Some Hostels and Hotels won't let you in with your Dog. Also be prepared to be appalled at the general cruel treatment of animals. Lots of strays.
It's heart breaking.
Suerte!