You can also gasify woody biomass into syngas, which can be used for energy production. The gasification process can be optimized for biochar (carbon) production which can be introduced into the soil (terra preta) for nutrient absorption and carbon sequestration. It's estimated that biochar is inert in the soil for approximately 10,000 years. So reverse mining of carbon.
You seem to be suggesting extracting the hydrogen to use as fuel, leaving behind the carbon in solid form. Instead of just burying it, it might be more usefully made into fibers to mix into and strengthen concrete.
Yes, this is also an option, although biologically derived char-ash is more than just pure carbon, there are a number of other minerals (calcium, magnesium, silica) that would normally result in ash locked up with the carbon. It would not be impossible to remove these, in a separate step, either before or after gasification.
Note, pure hydrogen is not produced in Syngas, it is instead a combination of Hydrogen, Carbon Monoxide, Methane and other higher order burnable gases (if full pyrolysis and reduction is not completed). Not to mention inert gases such as Nitrogen, Argon, etc. if air is used during the partial combustion phase.
All that being said, the biochar that is left behind is super useful as an additive to soil, as it acts as a sponge for nutrients, a matrix for keeping beneficial bacteria alive, and often contains volatiles that stimulate plant growth (as it mimics the volatiles given off after a forest fire).
