If I uninstall Bitcoin Core desktop client, what happens to the wallet.dat? Is this erased or left in place? If I install Bitcore Core desktop again, does my wallet.dat get overwritten?
Nothing happens to it. None of Bitcoin Core's data (which includes your wallet files, the block files, the block indexes, the chainstate, etc.) is removed when Bitcoin Core is uninstalled. All of this data is stored in a separate location from the installation.