Would dual screen monitors work if one monitor is connected to HDMI and the other is DVI on the same GPU? I ask because they come with good built in speakers, so I want to use the hdmi one for sound (only has 1 hdmi, 2 dvi on gpu). Unless is there a cord that I could connect from PC to the monitors to output sound without the hdmi?
Yes, that will work. You do not need to use the same type of connector. In fact, triple-monitor setups with ATI 5xxx series cards must use differing connectors (at least one DisplayPort must be used for a three-monitor setup).
Depending on your motherboard and the monitors, you could connect regular component output or digital optical from the motherboard to the monitor.