They report both.

In the case of Food/Energy, they're often very volatile, so sometimes it's useful to subtract them out. Basically, they're very 'noisy'- you'll have big jumps up and down depending on supply/demand, but that can obscure the actual trend. For example, you might have a spike in energy prices one month because a bunch of oil rigs were down, but they'll be back up and running next month. So it doesn't necessarily tell you anything useful to include that one month spike. When you subtract off Food/Energy, this is "core" CPI. But there is also 'normal' CPI which does include Food/Energy.

In the case of housing, those are considered assets. CPI measures consumption. They do include the 'consumption' part of housing, basically measuring what you would've paid in rent. So they try to basically split the consumption part (which is counted) from the asset part (not counted)