There is no downside to putting the grout just between the whole tiles, it's really just a personal aesthetic choice. The glaze does go over the entire tile, so if you don't grout all over it, nothing bad will happen -- the tile is still "sealed" with the glaze-- it's not like water will soak in or something. It's just that one way of doing it will look more like a grid of textured tiles and the other will look like many individual little pieces -- more like a mosaic.

What's the tile material and what is the application? A grout line is a personal decision, assuming all things are equal.

Yes if the tiles are not cracked and are in usable condition. You can also choose to leave minimum gap between two tiles.

Balaji, Try Burl Ywood from Roff. Even this color is not looking very odd. Please be aware that if you want to get grouting changed it is a troublesome process. If it is cement based grout you can get it touched up by an appropriate paint. Thanks Nitin