provides the easiest way to answer the two questions by providing two maps that show antipodal points on the Earth's surface overlaid on one another.

The coast of Colombia, South America seems to be antipodal to a stretch of the Sumatran coast in Indonesia. Perhaps Ecuador might be more ideal: near 80°W on the equator, antipodal to 100°E on the equator (still Sumatra). The antipodes to Peru's coastline seem to cross the coasts of Cambodia and Vietnam, while the antipodes to the coast of northern Chile intersects the coast of China. A point on the coast of Argentina is antipodal to a point on the coast of China just south of where the Korean peninsula juts out.

A point on the South American coast on the boarder between Venezuela and Guyana looks Antipodal to a point on the coast of Java.

Finding antipodal points that are as remote from coasts as possible is harder.  If done, though, it's best to get one point on land and the other in the sea. Perhaps central Africa, say 50° East 10° North, antipodal to 130° West 10° South in the Pacific Ocean.

Of course, the most sensible would be not to worry about coastlines and put the West pole at 90°W on the equator, and the East pole 90°E on the equator, so each is at an opposite center of the great circle formed by the prime meridian and 180th meridian (mostly the International Date Line). In a sense they really are the West and East poles, being the centers of the Western and Eastern Hemispheres.