How you need to place things depends on how you distorted the globe to get it to lay flat as a map. The different ways of doing this are called "projections".

Since you have a rectangular map, and it sounds like you want it to cover the whole globe, that means you are probably after some form of cylindrical projection. Cylindrical projections work by taking each parallel and then stretching or squashing so they all have the same length. The pair of parallels (one each north and south of the equator) that are not squashed or stretched are called "standard parallels".

The poles are also stretched out to form the top and bottom of the map, but you need to keep in mind that although they are lines on the map, they are points on the globe. If a line, say a coastline, touches the top or bottom, then it means that line passes exactly through the pole. The distortion at the poles is essentially infinite. That means that if the pole is inside a landmass, the coastline of that landmass should span across the map.

Where all normal cylindrical projections distort east-west the same way, they differ in how they distort north south. Equidistant Cylindrical doesn't change north-south distance (it matches up with the east-west distance at the standard parallels). Mercator adds north-south stretch to match the east west stretch, so it keeps shapes roughly the same, but distorts areas. Cylindrical Equal-Area does the opposite, distorting shape even more, but keeping areas the same. Since the distortion at the poles is infinite, Mercator adds infinite north-south stretch at the poles, which means the map is infinitely tall and only a finite portion of the middle of the map can be shown. In practice, Mercator is the only one of the three that is really suitable for a finished map. Equidistant cylindrical is mathematically simple which makes it useful for feeding into software to convert to another projection. Cylindrical equal area projections are pretty much worthless as there are far better equal area projections that are not cylindrical. Mercator was intended for marine navigation, with sailing ships, and for that purpose, it works reasonably well. It ended up getting used as the standard projection for reference maps for a while, but it is not particularly useful in that role and has been replaced with more modern projections like Winkel Tripel and Robinson. Mercator is also used for zoomable web maps like Google Maps and OpenStreetMap where its shape preserving property means that zoomed in sections are approximately correct, at least if you zoom in to a fairly large scale.

To get your currents right, you need to set up "gyres". The currents run arround in loops, going one way in the northern hemisphere, and the other in the southern hemisphere (sometimes with smaller gyres driven backwards by the bigger gyres like the Gulf of Alaska driven by the North Pacific Gyre.). Where the currents flow away from the equator, they are warm, and toward the equator, they are cold.