Yes, fossil fuels are originally from organic life, which resulted directly or indirectly from photosynthesis.

Plants do their best to suck up all the CO2 in the atmosphere. Yet CO2 is being added to the atmosphere faster than photosynthesis can capture it. This is why CO2 levels are rising. Human activity contributes a lot to the addition of CO2 (and equivalent greenhouse gases) and plants can't keep up with us.

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Yes, but fossile carbon is out of the carbon cycle, stored underground until we dig it up? This is a fairly fundamental pillar in the discussion around environmental impact of GHG emissions?

Yes, captured and stored at a time when there was many times as much CO2 in the atmosphere. Release it, and you return the atmosphere to its former condition.