Solution 1:

Partitions on internal drives in your computer are not mounted automatically, except for the partitions that were included in fstab. By default nowadays, only one partition is mounted in /etc/fstab, the root partition.

Partitions on removable drives (e.g. USB drives) are automatically mounted by the graphical desktop under /media/<username>/<string>, where <string> is the volume label, or the partition's UUId if the volume has no label.

How come the 2TB SSD is not listed in the fstab file?

See above. You still can have it mounted automatically by using the utility "Disks", or adding an entry in /etc/fstab yourself.

I suppose that a command gets executed during startup to mount the drive, where can I see that command?

External drives are mounted when logging in using the udev system.

The main screen shows the mount point as: /media/erik/0d210810-0cf4-48f3-ba54-47a08261c8e5 And the "Edit Mount Options" screen shows the mount point as: /mnt/0d210810-0cf4-48f3-ba54-47a08261c8e5. How come those two values are different?

The first mount point is how the drive currently is mounted through the udev system. The second mount point is how the Disks utility would include it in /etc/fstab. If you would activate that, then, next time, that would be the mount point rather than the first one.

My home directory is on the 256GB flash drive, how can I move it to the 2TB SSD?

By default, Ubuntu is installed with a single root partition, which also contains /home. One can install a system using a separate partition for /home. It is also possible to move the /home folder to a separate partition after installation.

How can I change the name of the directory at which the 2TB SSD is mounted?

Give the volume a label. That label will then be used by the udev system to determine the name of the mount point.