I may be off somewhat on some of the esoteric aspects (e.g., versioning), but here goes...
DVI is the transport, if you will, with 3 key flavors: DVI-A (analog), DVI-I (analog+digital) and DVI-D (digital). BTW, DVI was developed for connecting PC workstations to "dumb" digital flatpanel displays. DVI-D basically provides a digital raster dump to a display--uncompressed digital data--and is what *most* people mean when they talk about DVI in the CE world. DVI-D comes in single link (used today) and dual link (which can handle more data).
HDCP is the copy protection layer--it technically has nothing to do with DVI, but is used on top of DVI-D to enable the "evil" copy protection scheme. If you are using a DVI/HDCP-enabled STB coupled to a DVI/HDCP-enabled display, nothing to worry about. If you are using a DVI/HDCP-enabled STB coupled to a DVI display (without HDCP), and the HDCP flag is enabled for some specific content (e.g., HD PPV), something to worry about. In order to be 100% safe, you need HDCP on both sides of the signal chain. When people speak of DVI 1.0, they are talking about DVI/HDCP.
HDMI is simply the continued marriage of DVI and HDCP. As of now, at version 0.9 (I think), it is just a marketing campaign--single link DVI-D + HDCP with a new name. In the near future, HDMI will morph into dual link DVI-D + HDCP + digital audio, with a subsequent change in the connector to support up to 10 audio data channels. The new connector is going to be backwards compatible with the existing connector (likely via an intermediate plug or cable), so a future HDMI-compliant STB will still be able to connect to a DVI/HDCP display, albeit without routing the audio down the same cable. Why anyone interested in true HT would want their encoded audio sent to the display is beyond me...should be a big hit with people using HDMI-compliant TVs, though. HDMI will likely be the connection of choice for HD DVDs down the road--hopefully with analog grandfathered in for all of us who would rather not trash our existing HT setups.
It should be noted that the HDCP flag has not been enabled (except by "accident" on one digital cable system, IIRC) for any content as of yet, but even if/when it does get enabled, it is targeted at protecting things like HD PPV...not OTA and or semi-premium stuff like HBO-HD movies, etc. Time will tell, of course.