Since the world is round, flat maps (rectangular maps) distort the sizes of continents by making them larger than they actually are.
yes
Flat maps distort properties such as size, shape, distance, and direction of geographic features on the Earth's surface due to the challenge of representing a 3D spherical surface on a 2D plane. This distortion becomes more pronounced the further away from the equator you get.
DIRECTION
routes change all the time
All flat maps do, globes do not.
Relative population density maps, height elevation maps, and mineral deposit maps are all I can think of.
All of them. You pick a mathematical rule to go from the sphere to the plane depending on what you want to use the map for. Each rule distorts different things. There are 100's that have been used. Common rules do not conserve area. Others do not maintain shapes.
True shape maps, often referred to as conformal maps, preserve angles and local shapes but can distort size and area. These maps maintain the correct proportions of small areas, making them useful for navigation and detailed geological studies. However, as one moves away from the central meridian, the distortion of area increases, leading to inaccuracies in representing the true scale of larger regions. Examples of true shape maps include the Mercator projection, which is widely used for maritime navigation.
Flat maps distort Earth's surface because it is impossible to represent a curved, three-dimensional surface like the Earth's on a flat, two-dimensional map without some distortion occurring. Different map projections use different techniques to minimize distortion in certain areas, such as shape, area, distance, or direction, but there will always be some level of distortion present.
The proof that distortion occurs is to peel an orange in one piece, and then flatten the skin.
All map projections distort the surface in some fashion. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. There is no limit to the number of possible map projections.
Some other types of maps include thematic maps which focus on specific themes such as population distribution or climate patterns, topographic maps that show elevation and terrain features, and cartogram maps that distort the size of regions based on a specific variable like population or economic data.