I could be wrong but I think lawn tennis is just the old name for tennis - it dosen't necessarily have to be played on grass. Of course most clubs calling themselves lawn tennis clubs have grass courts - traditional name, traditional surface.
the name 'Lawn tennis' distinguishes it from real tennis (which is played in a walled indoor court).
'Clay court tennis' would normally be used in contrast to 'grass court tennis' not 'lawn tennis.'
Yes, you would play a game of tennis, not clay-court tennis. Unless you were talking about the professional circuit where a distinction is often made. Then you might say things like 'Nadal dominates clay-court tennis.' etc.