The following abbreviations are used:
AF = Africa
AS = Asia
AU = Australia
EU = Europe
NA = North America
SA = South America
Antarctica is not in the puzzle.
The only corner that has a border to its SE is A1. So D2 Germany tells you that A1 is AU and B2 is NA.
E5 Canada tells you that D4 and D5 are AU: Fiji and EU: Iceland or vice versa. Therefore, according to B5 Ecuador, AF: Malawi must go in E4.
Since D4 and D5 are AU: Fiji and EU: Iceland or vice versa (E2 New Zealand), B2 United States tells you that D4 is AU: Fiji and D5 is EU: Iceland.
D4 and D5 are AU: Fiji and EU: Iceland, so according to B5 Ecuador, either D3 or E3 is AU: New Zealand. But now you know from B2 United States that D3 cannot be AU, because it is directly south of EU: Germany. So E3 is AU: New Zealand.
Take a look at D4 Fiji. The equatorial countries that border India are A3, B3, and C3. The only way those three countries can each border two South American countries, yet only border three total South American countries, is if B3 itself is a South American country and A3 and C3 are not. So B3 is SA.
The only way for any country to border B3 Uruguay and another country with latitude 1 is if that country is C2. So, according to D5 Iceland, C2 is SA: Colombia.
According to D4 Fiji, either A2 or A4 is SA, certainly not A3. If A2 is SA, then B4 India would still need to be adjacent to another SA, according to C2 Colombia. Neither A4 nor C4 would work, for then B3 Uruguay would be adjacent to three SA countries, which contradicts D4 Fiji. According to E3 New Zealand, neither A5 nor C5 is SA either. So A2 must not be SA, meaning A4 is SA.
A4 Argentina tells you that B1 is either EU or AS. D5 Iceland told you that AF: Algeria has latitude 1 and borders C2 Colombia. E4 Malawi tells you that C1 and C3 are some combination of NA: Mexico (Canada is already placed) and AS: China/AS: Russia) This means that AF: Algeria must go in D1.
A1 Australia tells you that Longitude D contains countries from only three different continents. It already has three different continents, meaning that D3 must be either AF, AU, or EU. According to B2 United States, D3 cannot be either AU or EU, since it is below an EU and above an AU. Therefore D3 is AF.
Of the ten remaining boxes, only one can possibly border a NA country but not an AU country. Most of them obviously don’t work. E1 would work as AF: Niger if E2 were NA, but B3 Uruguay tells you that Longitude E cannot have another AF or another NA. The only place AF: Niger can possibly go, according to D3 Egypt, is A3.
At the moment the only possible places for two EU countries directly N/S of each other are C3+C4, C4+C5, and E1+E2. E1+E2 is out because B3 Uruguay says Longitude E contains countries from five different continents. C3+C4 is out because E4 Malawi tells you that C3 is either NA: Mexico or AS:China/Russia. So C4 and C5 must both be EU.
A4 Argentina told you that either A2 or B1 is AS, the other being EU. It could be one of the three that C5 Spain mentions. You know that C1 or C3 is AS: China/Russia, with the other being NA: Mexico. You know from B3 Uruguay that E1 and E2 cannot both be AS, if either of them even is. So with seven boxes remaining, you know that at least three of them are not AS: A2/B1, C1/C3, and E1/E2/both. So there are at most four remaining cells that can be AS. But now you know of at least four more AS countries in the puzzle: those mentioned in C5 Spain and AS:China/Russia in C1/C3. So now you know that one of E1/E2 does have to be AS. You also know that A5 is for sure AS.
B3 Uruguay tells you that Longitude A cannot represent five different continents. So A2 cannot be EU or NA. A4 Argentina told you that A2 is EU or AS, so it must be AS.
Since the three known corners are three different continents and there are six continents in the quiz, B1 Romania tells you that E1 must be the same continent as one of the other corners. B3 Uruguay also tells you that E1 cannot be a repeat from the same longitude, ruling out AU and NA. So E1 must be AS.
A5 Indonesia tells you that the number of EU countries = the number of SA countries. Right now the count is 5 EU and 4 SA. E4 Malawi told you that C1 and C3 are NA: Mexico and AS: Russia/China or vice versa. That means E2 has to be SA, to make the number of EU countries and SA countries equal, 5 each.
The countries that border eight countries are the ones that are not on the outside: nine total countries. All of them border C3 except C3 itself, but C3 borders B4 India, an Asian country. So if C3 is AS, every country that borders eight countries will border AS. This means, according to E3 New Zealand, that C3 is not AS. So E4 Malawi now tells you that C3 is NA: Mexico and that C1 is AS.