Does Obi-Wan retrieve his lightsaber from the end of Episode I for use in Episode II?
Nope. According to Master Replicas, the official lightsaber prop replica licensee, Obi-Wan has a new lightsaber in AOTC. His old lightsaber was destroyed in the Theed energy generator pit. Evidently, between TPM and AOTC, Obi-Wan built a new lightsaber very similar to his old one. Also, according to the novel Rogue Planet, 3 years after TPM, Obi-Wan used Qui-Gon’s lightsaber for some time after his death.