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In plasma cash, if Alice sends the same coin to both Bob and Charlie and they both try and cash out, who gets the coin?

Does the plasma sidechain have to periodically commit its plasma blocks to the mainchain to avoid this situation? If yes doesn't that mitigate all the supposed speed benefits we get from Plasma cash?

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So yes, the side chain does need to continuously post updates to the root chain. The sidechain submits the Merkle root of each plasma block to the root chain. When someone wants to withdraw his assets from the plasma contract he submits the transaction in which he received the asset as well as the Merkle proof for it.

In terms of speed, it does depend on how often the sidechain updates the root chain. Before the plasma block has been submitted, it is possible for a double spend to happen.

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