I know that on the 8th gen Corolla (1998-2002) - there are brackets that need to be transfered from the old deck to the new one, since the 7th and 8th gen are very similar - I assume the radio is the same way.
Most have threaded holes where the bracket is supposed to go - should be plug and go with the brackets. Wiring harness is nice thing to have (wiring looks nice, no splices needed) - keeps you from chopping up the existing harness, but not 100% necessary. The trim around the radio was designed for the OEM headunit. The aftermarket headunit will have a trim adapter in most cases - but I found that it doesn't quite flush with the rest of the dash.
A comprehesive dash kit would probably have the parts you need - but most of the time I find that they require some modification of the trim panel. Hard to say exactly for sure - chassis and front face design varys from unit to unit. One unit might look good without a trim kit, but another might look terrible. Have to just through the unit in there and see what it looks like and go from there.