I've never played with a joypad before, but people always seem to recommend joy2key for such things,

Yeah that worked for me, hero! Thankyou!

Oh yes! Works flawlessly!

* Go to Screamer 2's install folder
* Open "dosbox_screamer2.conf" with Notepad++ or some other equivalent (Windows Notepad sucks)
* Edit these lines and then save:
joysticktype=auto --> 4axis
timed=true --> false
* Double-click the 'Launch Settings' shortcut
* Edit one of control schemes by pressing/flicking the buttons/sticks
* Once you exit you will need to calibrate the controls. Just follow the onscreen instructions
* Done!

This method worked fine for me. Analog input is imperative while racing!

What about with Screamer 4x4? Does the 360 controller work with that without needing any extra software?

No idea. I don't own the game and I won't buy it in the foreseeable future. Fiddle around with it and maybe you will get it to work.
EDIT: As it was released in 2000 I don't think it needs Dosbox so the workaround won't work. Maybe it has a settings application like the Screamer 2 and hopefully it'll work right out of the box.
EDIT2: According to PCGW the game supports controllers but it's buggy.

Screamer 4X4 detects my wireless 360 controller (powered on before launching the game). In the controls menu axis and buttons can be assigned, except the d-pad. The calibration menu requests pushing all axis, checking center position before and after that.

Unfortunately there is no deadzone configuration, which makes it harder to drive straight for me.

The menus can not be controlled with the controller, or at least I don't know how. Axis can not be used as digitals, e.g. rear view, head left/right can not use the stick. Changing gear to reverse and back does not work reliably, some button presses are lost.