Hi Mike,
you plan to update the HMC, not upgrade. You can spare steps 2 and 3.
You need the HMC reboot before the update procedure only if you have never rebooted it, and want to be sure that it is in a "clean" state.
Upgrade data is not used during the update.
What is definitely the correct step is to save all LPAR profiles data before doing any HMC update or upgrade.
Another suggested step is to ensure that you have BMC connection configured to your HMC, and find the password for it.
Regarding firmware updates, you should usually plan the maintenance time for the procedure. You don't need to do it together with the HMC update, because the new HMC version supports your firmware. But your firmware is not the freshest in release 1050 (unless you mean 1050.50, which is not the same as 1050_058), and there is already release 1060. Meanwhile, there were several critical security problems.