Yes you can connect PS4 to DVI monitor. No convertor is needed, just a passive HDMI to DVI adapter. My nephews have done this to a 2006 era Dell DVI/VGA only monitor, we also have done this at work with several different DVI/DP only HP monitors. The only difference is it's a non-slim PS4.
Games always played and movies were a toss up. According to the Dell site, your DVI input supports HDCP. Switching the HDCP on/off in the PS4 menu didn't make a difference with the movies that wouldn't play. Honestly, it would play, but we'd get a signal lost on the monitor. We suspect, in retrospect, it's more to do that 24p output was enabled and the monitor didn't support it, but we didn't bother to go back and play around. They're just happy they can use it for games and YouTube casting.