Can I use both Mac mini display ports at the same time and which adapters do I need?
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.