PF: Is a researcher with a contract in a third country of exactly 12 months in the last 3 years eligible to apply for a Global Fellowship with an organisation in the same third country?

Yes, they are eligible to apply with an organisation in the same third country. However, if the candidate arrived in the country before the start date of the employment contract (e.g. to look for an apartment) or left after the end date of the employment contract, the period of residence starts from the day they arrived in the country and ends the day they left (e.g. dates of flight/ train tickets, visa, etc.).