Why Warrior lifted her propeller instead of folding it

Could HMS Warrior have used a folding propeller instead of hauling it up?

Not realistically: the necessary mechanisms and metallurgy were not mature enough, and a folding arrangement would have created reliability and battle-damage vulnerabilities.

A folding propeller required materials, control mechanisms, access, and watertight reliability beyond what the period could confidently provide. Its hinges and controls would have to operate through or near the hull without creating leaks or weak points. A warship also had to withstand enemy fire and battle shock rather than work only under peacetime conditions. Even a mechanically elegant device could be unacceptable if one hit disabled propulsion. Warrior's lifting arrangement reflects the difficulty of combining sail-era requirements with immature steam technology.