Synology products do allow you to expand volumes so multiple disks look like one drive. Not something I have done myself but I know it's possible.
Keep in mind that you won't be able to put your existing disks in to a Synology/Qnap/etc without the NAS wanting to wipe and prepare those disks for use in the array. Either putting in fresh disks and copying over or backing up somewhere else first would be an option in this type of scenario.
Finding my two bay 4GB NAS isn't enough so looking to upgrade too. The Synology DS 412+ is a better option than the 413J looking at the specs. 413J is a Marvel Kirkwood 1.6Ghz ARM based chip with 512MB memory and USB2 whereas the 412+ is an Intel Atom 2.13Ghz dual core with 1GB memory and USB 3. The speed difference in using it should be a lot better, worth spending that little extra imo.
The way I look at things is I don't need the extra speed raid gives and a lot of the stuff on my disks can be got again. So I have no raid and only backup/mirror a few crucial directories from one disk to another. If one disk goes down I still have another copy of my data. Gives me peace of mind as well as more space overall. I share certain folders on each disk which makes it more simple for me to index folders with XBMC and such like although I am at mercy of the problem you explained above in relation to some things taking more space.
All Synology products have the same DSM software, so looking at mine, I think it's Disk groups which is the part that lets you expand volumes
