Burkina Faso and Niger do share a border and are adjacent, though.
I solved it using the old green-cells-dont-contain-borderless-countries-hint from the comments.
I hope this is correct and I am guessing it can be improved:
F4 - Togo (only 4 letter country alphabetically after Russia)
F5 - Yemen (only 5 letter country alphabetically after Togo)
so we know row C has A-E-I-O countries (in case Y was considered a vowel)
C4 - Oman
E4 - Mali (only 4 letter country with capital alphabetically before Bangkok other than Iraq)
Emw - UAE (only multiword country with capital alphabetically before Bangkok)
Fmw - Vatican City (only multiword beginning with V)
C5 - Italy (only neighbour of Vatican so must be in the quiz)
E5 - Ghana (only 5 letter country with capital alphabetically before Bangkok other than Samoa that has no neighbours)
A4 - Peru (only 4 letter country left alphabetically after Myanmar)
C68 - Angola (there's no 4-5-8 letter countries, Antigua and Barbuda has no neighbours)
Cmw - El Salvador (the other multi-word E option was Equatorial Guinea but its neigbhours have C-G initials which are taken already) with Honduras as neighbour, somewhere in the first column
E68 - Jordan (other options with available letters are Brunei but can't have Malaysia, Greece but can't have Albania, Macedonia - actually now it's North Macedonia you could... - or Turkey)
This means S is Jordan's neighbour (can't have Israel or Iraq) either Syria or Saudi Arabia (in case Qatar has 'dubious' neighbour UAE).
F68 - Rwanda
This mean DRC is in the quiz as Rwanda's neighbour (Burundi is 7 letters long, U for Uganda is taken, T for Tanzania is taken)
B4 - Laos (only L countries possible are Libya but no cell is good for it because of row indications or green cell as it borders Chad; and Latvia but no neighbour can be present)
D68 - France: the only F country that can find a place; can't be in row A because alphabetically before Myanmar, cannot be in row B because capital is shorter than Vientiane
B68 - Honduras: only possible place for El Salvador's only possible neighbour
We have 6 letters missing: B K N Q S Z
Z is definitely taking the only 6-8 letters box left. K and Q are taking 2 of the 5 letter boxes as they have no multi-word answers.
B5 - Kenya: only place for 5 letter long country not bordering other countries in the quiz
D5 - Qatar: Doha is as long as Lima so Qatar can't go in A5, this means D5 is the only option
A5 - Niger: Nepal has no neighbours in the quiz so it's the only option
Bmw - Burkina Faso: Bosnia has no neighbours in the quiz, Burkina is alphabetically before Myanmar so this is the only option
A68 - Zambia: as Zimbabwe has no neighbours in the quiz
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