Words like “always” and “absolutely” are really pretty meaningless when using someone else’s server . I’ve been with DreamHost for almost 9 years now, and I have never “lost” email that does not mean that it could not happen.
The reason DH encourages you to keep a “tidy” inbox, is that many email clients automatically “load” the inbox upon connection (as does DH webmail) and with thousands of message stored there (for thousands of users), doing this creates a massive, and almost always unnecessary, load on the server. If once you have processed your inbox, you move message out of the inbox into another folder, those messages are only “loaded” upon request - which is much easier on the server, and everybody benefits.
IMAP is great if your routine does not lend itself to mainaging your mail via POP3; I still prefer POP3 with different computers and clients set to either leave mail on the server or POP it to my boxes, depending upon the circumstances, but others just “trust” the server. It’s really a matter of opinion.
Mail moved to another folder, whether automagically or manually, stays there until you remove it (depending upon your settings in the panel), but I would never interpret that to mean it is completely and absolutely safe on a shared server I don’t control any more than I would assume that any data on my own computers will never be lost.
The DH Terms of Service warn you of the “as is/as available” nature of the service, and specifically disclaim responsibility for data stored on their servers for these accounts. I trust them, and I’ve never lost data…but anything I absolutely need to keep, I back up to one (actually more than one!) of my own computers.