No, the 方 is playing the part of “whoever” here. Whichever (side) gets tired first loses (the game).
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The previous page is Sacchan doing both sides to give an example of how the game is played. (This is one of the cases where looking at a later page, then going back to the earlier page can give a false impression of who was speaking.)
The reason past tense is used for a game that has not started yet, I don’t know for certain. But here is my speculation:
The line is essentially, “The one who became tired first loses”.
However, Japanese tense doesn’t work quite the same as it does in English. Rather than thinking of た as being “past tense”, it’s more “completed action”. Thus, when the action of becoming tired has completed, that person loses.
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Almost caught up: I still have a few unanswered questions, maybe someone want to help me out:
Page 69
I am not quite sure what ののちゃん is saying here…
うーんと: ?? no idea
どういう: what kind of
こと: thing
かな: I wonder
I suppose it means What kind of thing is this, I wonder? Any suggestions for うーんと?
Thanks!
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Pg. 63
It is a form of ほら, taking the “there you are,” meaning. It is along the lines of どうぞ.
Pg. 69
I’m not 100% confident, but my interpretation was that it was うーん with the と quotation particle, and I think she was a little confused by the girls suddenly talking about throwing natto in the pond. So she was sorta saying, “(Sa-chan’s mom) said “No”… What kind of thing is this, I wonder?”
I think if I were to take liberties with the interpretation, a more natural English phrasing would be that she is saying, “I don’t get what you mean,” or “What’s going on?”
But I’m not totally sure of that, so if somebody has a more confident interpretation, please chime in!
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Glad to have confirmation of this! 
I suspected it, but couldn’t find a reference anywhere. Jisho lists ほれ as an alternative form though, which is close enough I suppose.
I took うーんと to just mean “umm” or something similar. She heard something crazy and she’s hesitating before answering as she’s still trying to process it. “Um, what are you talking about?”
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Yeah, I just read ほりぇ as a lazy/slurred pronunciation of that ほれ. Saito especially speaks in a lazy and slurred way in this manga, so my brain is properly primed for those kinds of odd slurring/spelling everytime I crack this manga open, at this point. 
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