Recently, I read that the Nambu-Goto action of a free relativistic string is motivated from the study of the relativistic point particle moving sweeping out a world-line parameterised by the proper time. May I ask

(i) Why do we parameterise the world-line of the particle by the proper time? Is it because we want to ensure that the relativistic point particle action remains Lorentz invariant? How important is that?

(ii) A free relativisitc string sweeps out a world-sheet that is parameterised by one time-like and one spatial parameter. Then we can write the action in flat space. But what happens when we have to work in a general curved space-time? How are we sure we are allowed to work with the flat metric, and all the results later (e.g. light-cone gauge quantisation, Virasoro algebra, etc) in flat space-time agrees with that of the general metric?

In other words, how can we assume that flat metric is equivalent to general metric? Has this got something to do with conformal field theory?

Thanks!