Can I use both Mac mini display ports at the same time and which adapters do I need? If I have two DVI monitors that are exactly alike, could I connect them both to the same Mac mini and extend the desktop using both displays? Can I use both the HDMI and Thunderbolt ports at the same time, and if so, what adapters do I need? 

Presumably, there is such a thing as a HDMI to DVI adapter, but what about Thunderbolt? I see that the connector of the Mini Display port to DVI adapter has the same shape as the Thunderbolt port, but will the Mini Display port to DVI adapter work? Thanks for the advice.
You can indeed use both the HDMI and Thunderbolt ports to drive two separate monitors with a Mac Mini.

Any Mac with a Thunderbolt port can use any mini Displayport display, cable or adapter, so you're correct that the mDP to DVI adapter is what you need1. If you have a display that requires dual-link DVI (typically only monitors with resolutions higher than 1920x1200), you'll need an active mDP to dual-link DVI adapter.

For the HDMI port, HDMI to DVI adapters are readily available (HDMI is essentially a superset of DVI with a different connector). Note that the HDMI port is limited to monitors with 1920x1200 or lower resolution.

1 Note that the reverse is not true - Thunderbolt monitors can't be used with mDP equipped Macs.