The most likely issue that prevents the user's mobile device from saving pictures to an external drive at home is that the peripherals are managed via MDM (mobile device management). MDM is a software solution that allows an organization to remotely control, secure, and enforce policies on mobile devices, such as smartphones, tablets, or laptops. MDM can also restrict the access and functionality of the peripherals, such as external drives, cameras, microphones, or printers, that are connected to the mobile devices.
One of the possible reasons why the user's mobile device can back up classified files to an external hard drive at work but not save pictures to an external drive at home is that the MDM policy allows only authorized peripherals to be used with the mobile device. For example, the MDM policy may require the external drive to have a certain encryption level, a specific serial number, or a valid certificate to be recognized by the mobile device. The external drive at work may meet these criteria, while the external drive at home may not.
Therefore, the mobile device can back up the classified files to the external drive at work, but it cannot save the pictures to the external drive at home.
To resolve this issue, the user may need to contact the IT administrator or the MDM provider and request permission to use the external drive at home. Alternatively, the user may need to use a different method to transfer the pictures from the mobile device to the external drive, such as using a cloud service, a wireless network, or a USB cable.
References:
* What is Mobile Device Management (MDM)?1
* How to Manage Mobile Devices With MDM2
* How to Connect an External Hard Drive to Your Phone3

The fuser is the part of a laser printer that uses heat and pressure to fuse the toner to the paper. If the fuser is defective or not hot enough, the toner may not adhere properly and smear when touched. The duplexing assembly, toner, and transfer belt are not related to this issue.
Reference: Official CompTIA learning resources CompTIA A+ Core 1 and Core 2, page 11.

Thunderbolt 3 is a USB-C port, so any device with a USB-C connector can be plugged into it. MiniUSB, microUSB and USB-A are not compatible with Thunderbolt 3.

802.11 ax is a wireless networking protocol that utilizes the 5.0GHz frequency and has the best speed and reception among the options given. 802.11 ax offers higher data rates, lower latency, increased capacity, and improved power efficiency than previous protocols. It is also known as Wi-Fi 6 or High Efficiency Wireless (HEW). 802.11a also uses the 5.0GHz frequency, but it is an older protocol that has lower speed and reception than 802.11 ax. 802.11b/g/n use the 2.4GHz frequency, which is more crowded and prone to interference than the 5.0GHz frequency. 802.11n can use both frequencies, but it is also slower and less efficient than 802.11 ax. Reference: https://www.comptia.org/training/books/a-core-1-220-1101-study-guide (page 82)

A router is a network device that connects multiple networks together and directs data packets to their intended destinations. This includes directing packets to networks outside of the local area network (LAN).
Routers use routing tables to determine the best path for data packets to travel.
Hubs and switches operate within a single LAN and are not capable of routing packets to external networks.
Bridges can connect two LANs together, but they do not have the intelligence to route packets to external networks.
Therefore, only routers are capable of directing packets to networks outside of the LAN.
References:
CompTIA A+ Certification Core 1 220-1101 Study Guide, Chapter 4: Networking, Section 4.1 Networking Fundamentals