On a discrete GPU, the card that has the GPU die on it also has the display output connectors - you could think of the card as being like a little motherboard; different card manufacturers can take the pins that come out of the GPU and route them to the phyiscal port connectors on the card. Different card manufacturers can choose different combinations of physical connectors - one may put DP + HDMI + VGA; maybe a different card vendor would choose DVI-D + DP, etc.
In the same way, the Intel GPU doesn't actually have any "fixed" set of physical connectors - those are chosen by the motherboard manufacturer (in desktop case) or by the laptop/tablet vendor as part of their overall system design. So... different motherboards/laptops will have various assortment of physical display connectors available.
That said, for DVI specifically, as far as I know, Intel display engine hardware only supports single link DVI so even if you connect a dual link cable somehow (directly or through dongles), you will only be able to get the max resolution that DVI single link supports. For NVidia, typically they would support DVI dual link.
If you want to drive high resolution displays from Intel graphics display engine, the best bet is Displayport (up to 3840x2160@60Hz) or HDMI 1.4 (with 1.4 capable monitor - up to 3840x2160@30Hz).
Be careful if you use dongles - passive dongles (which are cheaper) look to the gpu dispaly hardware like whatever is on the remote side of the dongle (no protocol conversion). If you could find an Active dongles for DP1.2->DVI-D, then you could use your DVI-D monitor at high resolutions driven off the DP port.
Thanks, I have learned a lot here! I guess all VGA monitors expect an analog signal, and therefore I have to get an "active" converter such as New DVI-D 24+1 to VGA Monitor HDTV Active Adapter Cable DVI2VGA Connector for PC | eBay or http://www.amazon.com/StarTech-com-DVI2VGAE-1920x1200-Adapter-Converter/dp/B00C93JQXS/ in order to connect a VGA adapter.
Oh, and will likely any DVI computer monitor you would buy these days work attached to the HD 4400 through a DVI-D (i.e. digital only) motherboard port, as in my case? or would some monitors being sold right now have a DVI port but yet expect an analog signal only? I guess digital should be the default for DVI monitors wouldn't it?