IMHO what you are more likely to see in "peripheral" areas is the attempt to settle local disputes between opposing countries who are not deeply tied to either alliance system. If the USA is busy in a conflict of this magnitude, as well as NATO, you might see South American countries go at each other to rectify border issues (Argentina-Chile, etc). I doubt any of these countries will attempt to go after NATO owned remnants (Falklands, French Guiana, Dutch holdings etc) unless and until it looks like NATO is seriously losing as poking NATO/USA in the eye could have negative consequences. The same sort of thing applies in Africa, Arabia/Iran, India/Pakistan where locals may decide to settle old scores. I can see South Africa letting NATO use air and naval facilities in prosecuting the war, and South Africa using its forces to deal with neighbors independently of what is going on elsewhere.
In the western hemisphere if Cuba and Nicaragua do anything except stay strictly neutral they will be curb stomped by the USA. The USSr may want them to do things like close the Panama Canal or attack oil rigs in the Gulf of Mexico. Attacking US soil directly would elicit a massive response. If the Cubans or Nicaraguans allow the USSR to base air or naval forces, that won't end well - they will pay a heavy price.
IMHO the vast majority of nations not in an alliance with the two blocs will not throw in one way or another unless they are forced to or if they see one side or the other as the inevitable victor and they want to get their share of the spoils. This refers to coming in on one side or the other with the big boys. As above, those not involved in alliances may very well start something locally that does not affect the major conflict. One South American country "readjusting" borders with another can be ignored for the moment, and dealt with when the big war is over. The key thing for those who are neutrals is doing whatever they can to avoid being involved in the big war unless and until they choose to do so.