The carpet was not been stapled or nailed under the lip of the step. Is this the normal way of laying the carpet of the stairs? If not, should I get the fitter back?
Not unless you specifically asked it to be done that way. The method you have is known as "waterfall" and is a standard way of doing it. Many people prefer it.
I too think the fitters done a great job. The other way seems more favoured in the US than over here. I've fitted quite a few as favours and not once has anyone asked for it to be done the other way ... just spoils the flow IMO
I think the 'bull nose' way seemed to be favoured by older people/generation. Waterfall way is the new preferred method.