I have two VGA connectors open on my monitor. Can I keep one DVI cable plugged in to my computer (as I only have one slot) and then plug the VGA cable into both of them? Will Windows 7 detect two monitors? (My computer doesn't have any HDMI ports)
VGA does not allow any kind of daisy chaining at all, so doing what you are proposing is not possible as described.
It is more likely that the dual VGA connectors are meant to be used as two separate inputs. In that respect you can connect both to the same system, and switch between inputs on the monitor to use the different parts of the larger screen as used by Windows (like you normally would with a dual-monitor setup, but with only one physical monitor).
VGA and DVI are not compatible, so you need some kind of active adapter to go from one to the other. To start with, VGA is analog, whereas DVI (like HDMI) is digital. DVI and HDMI are, however, electrically compatible but offer different feature sets.
Remember that VGA was introduced with the IBM PS/2 which was released to market in early 1987. For comparison, at that time, MS-DOS 3.3 was current, and Windows 2.0 wouldn't be released for another half year, and the text-mode brand-new OS/2 1.0 came out at the end of the year to take advantage of the powerful hardware; the most powerful model of the initial set (PS/2 Model 80, IBM model number 8580) sported an 80386 DX CPU.