What is the warranty period of these refurbished drives? Also is the warranty handled by Newegg or Seagate?

Factory recertified Seagate hard drives come with a three-month replacement warranty.

How is the packaging and shipping of this hard drive?

It was great for me. I've been running these drives 24/7 for months now, all have been good.

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Seagate (Recertified) Exos X 26TB Internal Hard Drive HDD - 3.5 In CMR SATA 6Gb/s, 7200 RPM, 512MB Cache, 2.5M MTBF (ST26000NM000C)

Factory recertified Seagate hard drives come with a three-month replacement warranty.
These drives are data sanitized, tested, and include all firmware and parts, like all Seagate factory recertified devices.
Market-leading HDD offering the highest capacity available for more petabytes per rack
Highly reliable performance making it the logical choice for cloud data center and massive scale-out data center applications
Hyperscale SATA model tuned for large data transfers and low latency
PowerBalance feature optimizes Watts/TB