Yep, DH is running Debian, so at least you will have the ability to have the same tools (if not exactly the same versions) available.
ssh, scp, or sftp should give you that level of security. If you are only worried about "man in the middle" versus actually protecting the content.
Of course you will have the challenge of determining when new file(s) have arrived in order to do the updates only as needed, but I'm thinking you can include that logic as part of the upload script, as DreamHost does allow remote MySQL connection (you can edit the MySQL user in the Control Panel -> Goodies -> Manage MySQL screen to specify remote hosts that my be used by that user to connect to a given database).
Both these issues should be manageable depending upon how you approach implementation.
How would DreamHost even know, and why would they care, whether the data transfer is scripted from your end or whether it is initiated by a user sitting at a keyboard?
I do feel it is worth mentioning that DreamHost, by the nature of it being a shared server hosting platform, is not well suited for running "extensive" automated scripting processes that are not directly related to the serving of pages. Their policies properly disallow the running of persistent processes, and their "procwatch" daemon is ever vigilant in tracking down and killing stuff that gobbles too much CPU/memory/etc.
While you can use "non-abusive" cron jobs for lightweight automation tasks, you can't run anything that is going to listen on a port (a server, etc.), but I don't see anything in what you have described that should required that. Really, I think you can script almost all of it on your own box and just automate the file transfer and MySQL updating.
I'd recommend doing all the scripting you can on your gentoo box (where you have greater control, can run persistent processes at will, etc.) rather than on DreamHost. If you just script your box to connect via ssh/sftp/scp/etc (your choice; scp seems simpler to me, but YMMV) you are halfway there.
To get the rest of the way there, just script your MySQL updating to use a remote MySQL connection and it seems to me you are home free. Of course, I know it's not that simple but that basic architecture seems to meet your needs and won't require any significant or extraordinary scripting at all on the DreamHost side.
Anyway, those are my thoughts on the subject. If any of that is helpful or useful at all then great! If not, well, you know the old saying about "free" advice being "worth exactly what you pay for it".
Good luck with your project. I think you will probably have little trouble once you get it sorted in your mind what to do on which box.