Does world size affect multiplayer lag?

No, but Yes and No.

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What is commonly referred to as 'lag' is the latency between client and server, or in other words, the correlation of the time it takes for a package (data) to be traded from the server to the client. This does not change when you have a bigger package (In this case: World Size)

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But Yes, while lag doesn't increase, bigger packages will take longer (there's more data to transfer, albeit the speed doesn't change. Same speed, more data = More Time).

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And no, the time isn't really going to increase. In theory the time does increase, however in practice, unless it's a ridiculously huge package, you won't even notice the transfer taking a milisecond longer

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In other words, having a smaller server will cause less time to be used to load things, but it's so small that it will barely be noticeable