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From what I understand almost all hash power on the network is used to secure the blockchain, but the processing power of the EVM remains constant even as the hash power of the network increases, and this is by design.

Why is this the case? Wouldn't it benefit everyone if the EVM could scale its power up as more powerful nodes enter the network?

To be clear I fully understand that the blockchain would grow a lot faster if these limits were not in place. Is that the only reason for those limits though?

Are there any plans for this to happen say, after Serenity and the introduction of Casper and PoS? It would seem to make sense to do that at that point since less processing power would be required to secure the blockchain, and nodes with lots of power could contribute that instead towards executing contracts. Is that a part of the plan?

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The "processing power" of the EVM is unrelated to mining hashrate. Power is not an accurate phrase for the EVM, because what limits computation on the EVM is cost: gas used multiplied by gas price. The Ethereum protocol does not provide limits on what the gas price and block gas limit can trend to. Heavy computations are possible on the EVM, but they consume plenty of gas, and leads to inexorbitant costs.

Lower gas prices and block gas limits that trend higher, are what allow for more computation to be performed on the EVM. The trend for both is relatively flat, because every full node has to perform every computation currently. As an open platform, where the reliability of the network depends on the number of nodes, it is desirable where anyone can set up a node and participate. Too many computations on the EVM would translate to long delays before a node is synchronized with the blockchain, higher CPU costs, and likely higher storage costs -- these would all decrease the number of full nodes participating.

As scalability improvements (such as sharding) and solutions are implemented and refined, it will help the situation where more computation can be performed on the EVM in aggregate, without significant burdens on nodes and compromising security. Block gas limits and gas prices may trend to allow contracts to perform more computations at lower costs. Improving scalability is an integral part of not just a plan, but ongoing Serenity work, as is Proof-of-Stake with Casper.

EDIT: "Processing power" of EVM increases with the imminent release of Homestead:

Block gas limit increased from 3141592 to 4712388 (~50% increase)
Default gas price reduced from 50 shannon to 20 shannon (~60% reduction)

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