In a sci-fi, post-apocalyptic scenario, is it reasonable/believable to assume that despite hardships, Humans would make time to engage in recreational sports activities? And in the case of sports' meetings/gatherings, like the ancient Panhellenic Games or the modern Olympiads, how would warfaring nations agree on where to meet, what rules to follow or whether there's fair competition - do these events not require a state of peacetime and international cooperation?
All depends on the depth of post-apocalypse
If civilization is still running well, we can see lavish, elaborate competitions like The Hunger Games. If the civilization is beaten back almost to the stone age, all we could see is The Blood of Heroes.
In either case, yes, sports is just in the nature of people. Even if they can't travel and can't get much of equipment, people will find ways to compete.
Alexander said it would depend on the depth of post-apocalypse; in most parts I agree, but the type of apocalypse matters as well: nuclear, zombie, alien invasion (like XCOM, maybe?), or otherworldly influence (a la Lovecraft, or perhaps like Half-life?)?
Nuclear Apocalypse-Humanity would likely survive a nuclear apocalypse just fine according to Anixx's comments, so sports will also survive and be played as normal within a short timeframe.
Zombie Apocalypse-The book series Last Kids on Earth suggests that teenagers would have video-game-style achievements and perhaps throw things at zombies for amusement. Hunting, football, hockey, soccer-if humanity survives, sports will too, and they will likely be turned against the zombies or played inside shelters.
Alien Invasion-Once again, look at hunting. In rebellion hideouts and shelters, even military complexes, other sports would likely have a place as a fitness maintenance program and natural entertainment. Perhaps certain humans will convince the aliens that they need to prove their supremacy in a fair and equal competition (just like in Space Jam) and play sports with them. Granted, said sports will be more than a little intense....
Alien Takeover-Humans in this case would be playing sports for alien entertainment, and perhaps even be "voluntold" for gladiatorial combat.
Otherworldly influence-No matter how strange the world becomes, people will still play sports (as long as they are still surviving, not necessarily thriving). Then it's just a matter of time until humanity adapts the current roles to fit their altered reality.
People will always find ways to keep their spirits up in the worst conditions, and playing games/sports would likely be a part of that. During the informal Christmas Truce of 1914, soldiers even played football with the men on the enemy side. Keeping morale up is essential to getting any kind of group through a crisis successfully. They're also useful for training people for physical fitness and working in a group- if you need to protect your settlement from raiders, playing lacrosse as war training might be a good idea.
On the other hand, international competitions and any other kind of event requiring complex logistics would be one of the first things to go in a societal collapse. So your post-apocalyptic sporting events would likely be smaller and more local. Another potential issue is fields and arenas being destroyed or repurposed (for arable land, emergency shelter, etc.), in which case sports would need to be modified to use other spaces.
TL:DR: People would still find ways to play sports after the collapse of civilization, but events like the World Cup would be off the table.
If you ask me, people would strive to make things look as much as possible to what was before the apocalypse.
This means, various bands of thugs as in many post-apocalyptic fictions are becoming less likely with time after the event.
More likely people will try to re-create the statehood, industry, armed forces, maybe democracy etc.
I do not expect massive violations of ethics in these games, like fights to death, gladiatorial combats, etc. Rather, the opposite, the people would strive to make such events as humane and hopeful as possible.
Also, even the worst apocalypse hardly can kick the humanity further than the 19th century technologically.