1. Can I add two more 2TB HDDs, and extend the existing RAID1 onto them?
The DNS-343 supports neither changing RAID type nor expanding the number of HDDs in a RAID without a complete reformat and loss of data.
2. Could I backup my existing RAID1 data, then delete the existing RAID1, install my 2 new HDDs, create a RAID1 across all four 2TB HDDs to deliver 4TB, restore my data? (may also have to recreate user&group permissions along the way)
From your post it looks like you are trying to create a RAID 10 configuration. The DNS-343 manual describes a "RAID 1 + JBOD" option. See the advanced configuration discussion in the DNS-343 manual. I haven't used this configuration myself, nor do I recollect any pertinent discussions on RAID 1 + JBOD on this board, so I can't comment on how well this configuration will work.
3. Some other suggestion from those out there who know far more than I do?
Conceptually, a RAID 1 + JBOD configuration should be more secure than a traditional RAID 10, since one failed HDD in one of the mirrors will not result in a complete data loss of the failed mirror. That said, many users on this board shared
unfortunate experiences with RAID implementations on this device, and IMHO creating a multi-layered disk configuration only compounds the issues that could arise if something goes wrong. When RAID works on the DNS-343, it works great, but when it breaks down, a number of owners have experienced catastrophic issues during rebuild attempts, while others have had no issue whatsoever. You'll have to weigh the pros and cons for yourself.
My one adamant piece of advice is that since RAID is NOT a backup, and you should maintain a separate physical backup of your data at all times. This strategy is not only prudent, but will ensure the security of your data in the event your RAID configuration breaks down.
Data Backup vs. Redundancy