Wheelbase
 

I've seen on setup sheets it looks like it's possible to widen the wheel base on the Team C tc02. But I can't see that I can do this on the Mad Rat itself?

I've fitted a 1mm metal washer between each wheel and hub which has widened the wheelbase by a couple of mm front and rear. It seems to have improved the handling quite a bit. (more than I was expecting actually)

Are there any downsides to doing it this way instead of buying the proper hex shaped spacers that you fit into the hex recess in the wheel?

Apparently there is "a box" at official brca events that the car has to fit in. If you widen too much it won't fit. I can't give any more advice than that but I'm sure those who know more about these things can advise

You are adjusting the "track" rather than the "wheelbase" and as Ashlandchris has said there is a BRCA limit.

I found that the front of the Team-C 2wd cars does not fit in the gauge box until you remove about 1mm from the end of the stub axles so if you have taken the wheel rim or tyre out past that point then it will not be legal.

At the rear, it depends which tyres you are running. I found that my TC02C was a good fit in the box when using the kit tyres. These overhang the wheel rims quite a bit so when you fit something like a Schumacher mini spike there is room to make the track a few mm wider. I made some aluminium hex drives to do this.

What sort of handling characteristics can be expected from a slightly wider rear end to front end or doesn't it make much difference?

If I had to guess, I'd say slightly better grip, make the suspension feel slightly softer and slightly better stability wrt grip roll etc.
But I'm no expert...