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Why the jump from "sports" to "organize international sports event"?
Is it not enough to play at your neighborhood hoop with your buddies?
Sports can be fun, organizing huge events is not, ask event organizers.
Imagine the stark contrast of a chill day of going out surfing vs calling people, scheduling, setting dates, booking hotels, etc. Much has been written about the corruption in FIFA, the slavery in preparation, the negative impacts Olympics have on the cities that host them, etc.
Also, if anarchists organized a sports event, you can be sure it would not have "national" in the premise.
This premise seems written by a communist trying to troll or fit in. I will answer al lthose questions to show how silly they are:

"Can anarchists - specifically meaning here anarchists who embrace the idea of egalitarianism through collectivism, communism, communalism etc., - play games that involve a discernible winner and loser though?"

Yes. They may even engage in dom and sub role play, shocking I know. A thing done for fun with consent among all parts involved is okay. It's not as if sports winners or doms will be declared supreme rulers outside of the court or the bedroom.

"In communist-anarchist snakes and ladders, do we not all move round the board at the same speed, so as to not leave anyone behind in the revolution (of going up and then back down and the back up and then back down, revolving again and again and again and again)?"

I don't know, please share the link where we might see the rules to this game. Or is this a shitty metaphor?

"If we accept the idea that within an anarchist society we would seek to reproduce every day normal narratives like public transport, agricultural projects and media projects, why would we not also include inter-national (or perhaps inter-communal) sports events, like the Olympic and Para-Olympic games?"

This alludes to a valid critique that anarchists should not reproduce every day normal narratives like those you mentioned which are encompassed in the big bads we know as civilization, patriarchy, states, capitalism, and all kinds of oppressive systems/constructs/hierarchies.

So I accept the implicit critique, but refuse it as base for an honest question.

"How would we differentiate winners and losers - or does everyone just get a participation trophy?"

Do you want us to explain how sports work, or ? Each sport, as certain types of games, have clear rules that determine the winners and losers. Prizes need not be a factor, but could be.

"What sports would we include, if we don't want to exclude sports that aren't popular within the consumer-media spectacle of mainstream games?"

What? Do you want us to examine which sports are trending? Dude, you can do sports. Who is banning you from doing sports? Where is the strawman that cancelled sports? I don't see it.
Are you lashing out against anarchists because sports events got cancelled or postponed because of coronavirus?

"Are cultural-events like sports tournaments incompatible with this vision of an anarchist society (or any anarchist-society)?"

Anything is possible...if you believe. All you need is to be decided.

"Are anarchists then forbidden to play sports, regardless of whether or not they want to enjoy the fun, out of the potential risk of creating a form of social-hierarchy that would contradict egalitarian principles?"

All is allowed, nothing is forbidden. Anarchists refuse to rule or be ruled, oppress or be oppress because they so desire it, not because someone imposed a rule on them telling them to be like that.
Caricaturesque egalitarian straitjackets are stuff of the moralistic authoritarian left and communists.

Have fun playing sports!