If anybody's interested, here's how to get it to work without an analog controller of any kind.
First you have to run the game as if you ran it for the first time, so if you already ran it, and the controls don't work, you have to remove the harddriv.nv file in the nvram directory.
Then start Mame and select "default game options". Under controllers, select "enable mouse input" and under controller mapping, select "mouse" as paddle device. Hit apply, then go back to the same menu and make sure what you did is still there.
Run "Hard Drivin'" and as soon as you see the "Initialize POT inputs", hit the TAB key and select "Inputs for this game".
Scroll down to "P2 Pedal 1 Analog" and hit RETURN, then move the mouse up/down or in the vertical direction. You should see "Mouse 1 Y" in the space next to it.
Make sure both "Paddle Analog" says "Mouse 1 X". If not, scroll over then, hit return and move the mouse horizontally left/right until you see "Mouse 1 X" next to both.
Now, hit TAB again to get rid of the key configuration screen. You should be back at "Initialize POT inputs" screen.
Hit the "1" key, which is your ignition key. You'll immediately see "Turn wheel counterclockwise then turn key". Move the mouse horizontally to the left and hit the "1" key. Then it will ask for "Turn wheel clockwise then turn key". Move the mouse horizontally to the right and hit "1". Your streering should be configured.
Next you'll see "Initialize shifter limits". You don't need the shifter because there is an automatic, so you can just skip this, but just in case, hit "D" + "R" keys at the same time, and while still holding them, press "1". Then on the next screen hold "F" + "G" and then hit "1".
Next, it will ask for "Seat Limits". This is obviously unnecessary, so hit "1" twice to get rid of it.
The very next screen is the root of the problem, so pay attention. When it says "Initialize Break limits". WAIT for a second or two, and when it asks "Step on the break and release", move the mouse DOWN then UP. It should recognize this, and say "thank you".
The game should be running now, but we have got the mouse linked to both streering and breaking, which is not good, so hit TAB again, and go back to "P2 Pedal 1 Analog". Hit RETURN over it then ESC. This will kill the analog break input. Then go down to "P2 Pedal 1 Inc", hit RET and then "Z". So you have the break in digial mode now, where "Z" hits the breaks and "A" lets off.
Press "5" to insert a coin, then move the mouse to the left for "Automatic transmission" then hit "1". You should be in the car now, so press and hold "1" again till it starts. "Left CTRL" is your gas pedal, so step on it, and if you hear the wheels screeching, hit "A" to let off the breaks. (THIS WAS THE WHOLE PROBLEM!).
Now be prepared for some real fun. I also recommend messing with game options again. You can actually make the game run faster than in the arcade by selecting some of the advanced timing options. If you got a supercomputer at home, you can fly through the game. Also, beware that "Race Drivin'" runs slower than "Hard Drivin", so don't use it.
OK, that's all my good friends, and god bless.