Trees that lose their leaves in winter are hardwoods, those that don't are softwoods. The density of the lumber has no bearing on the classification. Therefore pine are considered to be softwood.
No. Pine is not a hardwood. It should not be burned in a fireplace or wood stove because the flammable resins in the wood will build up in the chimney and could ignite causing a chimney fire.
beech is a hardwood. this means it is from a broad leaf tree which looses it's leaves in the autumn (or fall in USA). it does not mean that it is harder than wood from a conifer which does not loose...