Windows allocates drive letters when it loads. There is no way it will allocate the same drive letter to two drives or partitions.
When you re-attach your old HDD to the system, Windows will allocate the next available drive letter to it.
Letter 'C' will not be available because it's already in use by the SSD, so there is no way there will be a conflict.
And, yes, when installing Windows you should always disconnect any other hard drives that aren't required for the installation.