New SSD. After installing Windows 10, can I reinstall a Lenovo partition (with One-Key, etc.) from a backup on USB? (Lenovo 300)?

If you know how to install the One Key software onto a hard drive partition you will be able to do this

Perhaps. It depends on how that "Lenovo partition" was set up by the maker or you on the previous drive installation. Some have it only on the head of the active partition (where there's your new Windows 10 and the MBR, some other necessary files).
But high activity level programs should not be used on an SSD. Constant writing and deleting will degrade it rapidly. This will not happen on a spinning hard disk.

Lenovo one-key is used to factory restore to windows 7. As you have installed windows 10 Lenovo one-key is redundant as Windows 10 already has a reset feature.