I am new to Ethereum. I've just installed the Mist Wallet, and I find that it keeps the CPU running HOT. Is the wallet a node on the Ethereum Network? If so, should I be able to mine stuff?
If you run the Mist wallet you run a node.
I find that it keeps the CPU running HOT
The reason should be that your wallet is syncing to the blockchain.
should I be able to mine stuff?
I've made my small research and I haven't found any proofs that you can mine using the Mist Wallet and have a reward for your work. People suggest connecting to mining pools and using software like Claymore or Ethermine if you actually want to mine and be rewarded.