I just got a new Logitech Chillstream (to replace the old one after a friend got sick on it after a binge). Win7 recognizes it as am Xbox360 controller, which it closely resembles. I can get Cloanto's Amiga Forever to recognize it (again, it shows up in the WinUAE settings as XUSB Controller) and most games that have an Xbox analog have their own setups for Xbox controllers, but I'm having some trouble getting games run in DosBox to recognize it (I use the D-Fend Reloaded frontend). Is there a way to make Dos games see it as a Gravis Gamepad or something?

Are there any options in the software you got that have the word 'legacy' in them? You might be able to set up something using xpadder but I think that's not freeware (don't own a pc gamepad so have never needed to do any of this).
Found a link that implies a custom driver will work for dosbox stuff
http://i.org.helsinki.fi/lassial/guides/100327-Play_Tie_Fighter_in_Windows

Getting controller settings to work correctly in DOSBox can be very tricky because you need to set a suitable one in DOSBox as well as in the game itself, and for earlier games you're limited in what buttons you can actually use.
I ended up ignoring DOSBox's own controller support, using Xpadder to redirect the buttons to keyboard/mouse inputs. Despite its name it works with all controllers, not just Xbox-like ones. This is the last free version of Xpadder; newer versions cost money but are mostly identical to this one. Setting up a template for your controller can be a bit fiddly but it only has to be done once per controller and after that it's really simple.
EDIT: if you go this route you will want to disable controller support in DOSBox entirely so that the game doesn't see conflicting inputs.

Depending on the game, wouldn't it be better to map the four analog axes and four buttons offered by the old analog gameport interface to the gamepad as desired, then fill in the rest of the controls with Xpadder or a similar utility (making sure to leave the controls bound to the emulated joystick unused)?
Besides, some games (like the aforementioned TIE Fighter) REQUIRE a joystick to play anyway, as in through the gameport interface.
I know it's a bit more hassle this way, but it'll undoubtedly let you have analog control in games that allow for it.

Actually TIE Fighter doesn't require a joystick. One time when mine broke I spent a fortnight playing it with a keyboard & mouse, naturally I flew nearly as badly as the "real" TIE pilots which makes me suspect the Emperor was a cheapskate bastard who refused to buy his pilots decent equipment (explains the lack of shields too)

Seconding Xpadder recommendation. It's very good in these situations.

Hmm, ok, so I can use Xpadder so bind keyboard keys to the pad's controls, but there's some problems it doesn't cover. For instance, when I tried to calibrate the d-pad and sticks through Control Panel, it seems to think that the two secondary L and R triggers are the Z-Axis, and that the D-pad is a Hat Switch, and that the right analog stick is the X and Y Rotation. It doesn't make sense because it counts the analog stick buttons (you know, when you push them straight down, into the controller) as actual buttons, and you'd think it'd count THOSE as the Z-Axis instead of L2 And R2.

Well clicking the sticks is just a digital control, you can only press them all the way or nothing unlike the triggers

This is by design, and works exactly the same with a real 360 controller.
The triggers are an axis to provide analog sensitivity for throttle-style input in games that support that (even many older games recognise them for this purpose, mostly racing/simulation-style games). Because the assignments fit legacy expectations older games will usually see all of the controls, although some may have problems with the triggers depending on how they supported joystick throttle axes back in the day (some see them as a single button, others don't see them at all) and some may see only one or neither of the right analog axes for the same reason. These are technical limitations of the games themselves, not your particular controller.
Xpadder doesn't require you to map all controls at once, so one possibility is to use the game's own controller support for all inputs it sees and then use Xpadder for the remaining ones it can't see.
EDIT: these detection problems only apply to older games, of course; new games designed with 360 controllers in mind will see everything, and since Windows itself sees your controller as a real 360 controller you shouldn't have any problems. All Valve games and most Games for Windows games fully support 360 controllers and will make a note of this on the box/store entry/etc.