No no no no. Is replacing the drop ceiling with drywall an option?

You might get another foot of ceiling that way. For sure, painting the loose pieces and the dividers sounds like a nightmare...

Drywall and framing to get around plumbing and hvac is extensive in a basement. It can be a great improvement.

Otherwise, use a sprayer to paint it in place. Rolling would be difficult. I would not disassemble your ceiling unless/until you do drywall. Color is your call, but I would stay light.