Will this radiator work with the hybrid engine?

No — this is calibrated for the 2.4L non-hybrid model only. Hybrid vehicles have a different cooling circuit. Match your engine size and drivetrain before ordering.

Do I need to drain the cooling system before swapping this in?

Yes. Drain your current coolant completely, then remove the old radiator at the factory mounting points. New radiators ship empty — fill and bleed according to your service manual once installed.

What hoses or gaskets come with this?

The radiator assembly ships as-is. Your existing hose clamps and gaskets can be reused if they're in good condition; inspect them during installation. Coolant is not included.