Is it possible to work outside of the US on a K1 fiance Visa until you are allowed to work (receive I-765) in the United States? Or do I have to stay in the United states for that time?
There are two issues here:
K-1 is a single-entry visa, so you will not have a valid U.S. visa after entering. So if you leave the U.S. you will need a new visa (paying the fee again) to enter. (Plus, if you're married at that point, you won't qualify for a K-1 visa, so the only way to immigrate would be through Consular Processing abroad.) The only exception is travel of less than 30 days to Canada or Mexico within the 90-day I-94 validity period, after which you can re-enter the U.S. with automatic revalidation.
If you leave the U.S. while Adjustment of Status (I-485) is pending, you will automatically abandon the Adjustment of Status (which would be a huge waste of money and effort) unless you have H, L, or K-3/K-4 status (which you don't), or you have Advance Parole. From your description ("waiting for green card"), it seems like you have already applied for Adjustment of Status. You can apply for Advance Parole at the same time as applying for Adjustment of Status, and if you also applied for an EAD (I-765) based on AOS, then the EAD and AP (I-131) will usually come together on the same card. They are both free, and they both take almost 90 days to get. After you have Advance Parole, you are free to travel out of the U.S. while waiting for the Adjustment of Status to finish. (However, by that time you will also have an EAD, which will obviate your reason for wanting to leave the U.S. in the first place.)
From the USCIS:
After admission, your fiancé(e) may immediately apply for permission to work by filing a Form I-765, Application for Employment Authorization with the USCIS Service Center having jurisdiction over your place of residence. Any work authorization based on a nonimmigrant fiancé (e) visa would be valid for only 90 days after entry. However, your fiancé (e) would also be eligible to apply for an extended work authorization at the same time as he or she files for permanent residence. In this case, your fiancé(e) would file Form I-765 together with Form I-485 as soon as you marry.
Based on that you should be able to obtain employment authorization as soon as you enter the country and continue it until you receive the Green Card.
No you can not. This is because AFTER arrival on a k-1 visa you may not depart the USA until AFTER either your green card is approved and issued, or until you are granted Advance Parole. Advance Parole provides you with specific permission to exit/enter while the green card is in process. When applying for Advance Parole, you ALSO have the option to apply for Employment Authorization. Employment Authorization allows you to work while the green card is in process. Both employment authorization and advance parole are granted the SAME time. In fact a single card is issued for both. This means the EARLIEST date you are allowed to leave (advance parole) is ALSO the earliest date you are eligible to work,