Textures aren't going to survive reprojection anyway. Keep your symbology simple and don't waste time on decoration or careful label placement until you have the map in its final projection.

Making the top and bottom line up would give you a toroidal topology, not spherical. I assume your world is not doughnut shaped.

Normal cylindrical projections like this stretch out the poles from single points to lines as long as the equator. All the parallels in between also get stretched to the same length. So you get a varying amount of east-west stretch as you move north and south. In this particular projection there's no corresponding variation in north-south stretch. You need to draw that varying east-west stretching into your map or it won't look right when reprojected.