This is more of a question about your OS and hardware other than armory in specific. A full node needs to write the blockchain to a drive that your computer recognizes.
So long as your computer can recognize your drive you could do anything with it.
It doesn't matter if the drive is in or out of the box for that matter (external vs. internal). Just set the wallet to store the blockchain there and have have connected before starting your client.
Not exactly. While Core only does plain read/write operations to the drive, Armory maps the files for access. This means it tells the OS to treat specific areas of the drive as part of the RAM. The OS then balances that stuff in and out of the RAM according to access patterns for significantly better performances. The caveat is that stuff like NFS won't work. Physically connected drives will however, through USB or internally.
Additionally, external drives are generally a lot slower than internal drives due to limitations of the interfaces used for this ie. USB... that may impact on your initial blockchain syncing time (which involves a lot of disk I/O)... but is probably not so much of an issue in day to day usage.
If it's a desktop computer, a 1TB hard drive is a lot cheaper than external drives with the same capacity.
Plus it will be faster since it's internally connected.
That's not necessarely true. NVMe drives are fast because they connect directly through PCIe. USB 3 is slow not because of the transport capabilities but because it typically goes through the southbridge. I think some manufacturers connect some USB-C ports to PCIe directly but that really depends on your mobo. If you straight up buy a PCIe USB-C adapter card however, you can get blazing fast speeds with a USB-C to m.2 connector.
That's not particularly relevant for a HDD, which is limited to ~100MB/s sequential I/O cause of the physical constraints, but you can get near native speeds with an NVMe drive plugged through USB-C that way. USB-C can carry 4K@60fps video for what its worth.
