The other airline flying from UIO to the Galapagos is Aerogal, both Aerogal and Tame stop briefly in Guayaquil (in case you want to join the flight there).

You won't be able to use your miles on these airlines, for BA you can only book with their partner airlines, same for AA (partner airlines means oneworld plus a couple of others).

You'll have enough miles by the time you get to NZ/Oz to get a flight, however jetstar and virgin blue are very cheap there so don't go wasting miles on a cheap flight.

Also don't go out of your way to get miles, you can buy them for 2.5c each so don't undertake a transactions, e.g. an expensive (annual fees, fx trans fees) credit card unless the miles you'll get are going to be significantly cheaper than 2.5c.

Regarding the SCL-AKL availability you can use www.seatcounter.com
It will show LA but doesn't seem to work properly for BA flights. When we booked this route it was full for every day up to about 3 months out, however this depends on the time of year days can open up due to cancellations, etc.