Naturally in English, as in other languages, certain words have many synonyms -- many of which as similar, but not necessarily the same. The best way to learn the exact definitions is to notice what things the words are used with.
"Flat" refers to a surface that is generally planar (forming a plane in space), in that it does not curve. A table is usually flat, a wall can be flat, and a television screen (nowadays) is flat. An angled surface may be flat as long as it does not curve up or down.
Sometimes flat is used to mean horizontal as in "lie flat", but a horizontal surface is not necessarily a flat surface.
"Level", on the other hand, implies flatness but usually means perpendicular to the force of gravity. A level surface is one which does not significantly curve up or down from side to side. A tennis court is level (and also flat) because one side is not higher than the other. A building should be constructed so that all the horizontal beams are level. Something that is balanced over a fulcrum will be level.
"Smooth" relates to the surface quality. A flat surface can be rough (bumpy to the touch) or smooth -- but if it is smooth and level, we might call it perfectly flat. The surface of a ping-pong ball is smooth. Someone's skin might be said to be smooth (if it is free from blemish). An ice skating rink will be relatively smooth, as well as flat and level.
"Even" is a little more general. An even surface might be level, or smooth, or flat. Even simply refers to the fact that the object appears similar from one side to the other If you compare different parts of the surface there should be no marks or imperfections or significant differences in one part that there are not in another. The surface doesn't have to be smooth, but if not, the texture should be regular.
An example of an even surface is a grassy sports field. The grass makes it a rough surface, but there are minimal differences between one part of the surface and another.