Here are the meanings of the operators:
The operation X # Y means concatenate X+Y into XY.
AUSTRIA # BENIN = AUSTRIABENIN
The operation X & Y means that, for each letter occurrence in Y, remove one occurrence of that letter from X if possible.
AUSTRIABENIN & UZBEKISTAN = RAIN
The operation X = Y means that Y is a nation and is an anagram of X.
RAIN = IRAN
Solved Questions
1. Iceland # Iran & China = Ireland
2. Canada # Lithuania & Nicaragua = Thailand
3. Austria # Slovenia & Elsavador = Tunisia
4. Slovenia # Greece & Cote D Ivoire = Senegal
5. Canada # Mongolia & Macedonia = Angola
6. Mauritius # Tanzania & Uzbekistan = Mauritania
7. Germany # Romania & Singapore = Myanmar
8. Germany # USA & Timor Leste = Guyana
9. Lithuania # Canada & Nicaragua = Thailand
10. Algeria # Sudan & Israel = Uganda
11. Austria # Malawi & Zimbabwe = Australia
12. Mauritania # USA & Canada = Mauritius
13. Thailand # Ukraine & Denmark = Lithuania
14. Indonesia # Jamaica & Tajikistan = Macedonia
15. North Korea # USA & Iran = South Korea
16. Estonia # India & Italy = Indonesia
17. Australia # Congo & Lesotho = Nicaragua
18. Armenia # Australia & Elsavador = Mauritania
19. Estonia # Nepal & Jordan = Palestine
20. Honduras # Tunisia & Nigeria = South Sudan
Bonus
Tonga # Laos & Sri Lanka = Togo
Russia # Nepal & Belarus = Spain (<- credit to Jonathan Allan)