What happens if there are no transactions in a block? Suppose there is a 10 minute time frame in the middle of the night, maybe on a holiday, and there are no transactions made. Do the miners get paid 50 bitcoins for processing a zero?
The mining reward must be collected with a transaction, so technically there is at least one transaction in each block.
The mining reward was designed to enable the initial distribution of bitcoins and to encourage the creation of a powerful bitcoin network with a lot of computers to support it, therefore it makes perfect sense that a miner gets the block mining reward regardless of how many transactions were included.
The answer to your question is yes. The transaction fees reward miners from processing transactions. The block reward is for securing the hash chain. They're not "processing a zero", they're securing every transaction made in every previous block, increasing the number of confirmations it has and increasing the difficulty of undoing it.
It doesn't required there be zero transaction in a 10 minute period. Each miner (or pool) decides which transactions to include and they can choose to include nothing.
However all blocks will have at least one transaction the coinbase transaction which includes the signer's reward address. There must be one and only one coinbase transaction for a block to be valid.
The coinbase is where new coins are mined out of thin air to increase the money supply. This transaction also transfers any transaction fees to the miner. If block reward (subsidy) and fees are both zero then then this transaction will still exist it will just have a value of zero.