That doesn't seem to be true. Like an economy that doesn't funnel money into individuals. Or even moneyless economies like Library or Gift. (Though moneyless economies imply we're achieving actual communism, going beyond socialism)

No, socialist systems like free housing, healthcare, education can exist alongside capitalism. Worker owned systems like cooperatives still operate in a market.

Are you talking about free housing (etc) programs being managed as a cooperative, alongside a commodities market of cooperatives? If yes, that's not capitalism, that's socialism. If no, then you must be talking about a welfare state like what's in Scandinavia, which isn't socialist.

Kind of relevant to both points, there are a few different schools of socialism so you could see if any make more sense to you.