(1) will the mere installation of the SSD, with all its Windows 7 files, somehow inadvertently corrupt the OS in the new computer?
If you keep the Windows 10 drive as the boot drive, no.
(2) what is the best way to remove all the files in the SSD after installation in the new computer and thus have a clean SSD available for extra storage?
You can see the Windows built-in Disk Management app in action at https://www.computerhope.com/issues/ch001765.htm or you can use the command line (steps follow).
The DISKPART command which is standard in Windows 10.
A) From Windows 10, launch a Command Prompt WIN-R CMD Enter
B) In that window, run
C) In diskpart, do
list disk and you should see your boot drive as Disk 0 and your newly added drive as Disk 1. If you don't, stop, go no further, and consult here!
DISKPART> list disks
Microsoft DiskPart version 10.0.16299.15
DISK - Display a list of disks. For example, LIST DISK.
PARTITION - Display a list of partitions on the selected disk.
For example, LIST PARTITION.
VOLUME - Display a list of volumes. For example, LIST VOLUME.
VDISK - Displays a list of virtual disks.
DISKPART> list disk
Disk ### Status Size Free Dyn Gpt
-------- ------------- ------- ------- --- ---
Disk 0 Online XXX GB 0 B *
Disk 1 Online YYY GB 0 B
sel disk 1 to pick your newly added old Win 7 drive.
del partition 1 to remove the first partition.
list part to check for any other partitions; if there are others, delete them, too, as in step E above.
G) Once all partitions on that drive are gone
create partition primary
FORMAT FS=NTFS LABEL="DATA" QUICK to format it (you can change DATA to some other name if you wish).
ASSIGN to have Windows assign the next available drive letter.
J) Quit with
K) Check in Explorer for your new empty drive.