It depends on your object of classification. If color has no significance in your images to classify then its better to go for grey scale images to avoid false classification and complexities.
Hi there. Well, maybe you could try both at the first insight... It depends on many things. Remember that a RGB image has 3 dimensions and grayscale has just one, so, everything tend to be more costly, but if it brings better results, go for it. Sometimes other colospaces (or color map) like HSV, L*a*b* etc., can bring other kind of information.
Depending on what it is that you are wanting to classify, don't forget that non-visible spectra can also provide diagnostics above either RGB or monochrome imagery. The resulting images may be either monochrome or false-coloured, but the filters / tech used in producing the images is what will help to differentiate the classification.
That depends on the nature of images. For example if you want to classify some satellite image then the more bands will help you more as they have more information.