The simplest way to get the correct projection is likely to just draw directly on the correct projection template (but I think you know that). For minimal distortion, an interrupted projection would be best, but the interruptions would need to be placed so that the splits avoid land for best effect. Without knowing where your land masses are, it's hard to design a final set of projections that minimizes distortion. Much of your areas of interest are in the equatorial regions, though, which means that a reasonable whole-world compromise projection will probably do well enough.
The Hammer projection is a relatively pleasing-looking compromise projection. Mollweide is also popular because it doesn't have the curved parallels. Any projection will work, really, as long as you have an understanding of the distortions and can deal with them. For my convenience - strike that, your convenience - I've attached some basic 15 degree grid whole-world templates for Hammer and Mollweide projections.
Hammer2a.png Mollweide2a.png
Hammer1a.png Mollweide1a.png
The top row has 15 degrees lopped off each pole to avoid bunching up of the lines. The bottom line has the full ellipse for each projection. The outermost line on these is half the thickness of the others, which is an unfortunate artifact of the elderly piece of software that I used to do these.
A tool like
Flex Projector will let you get a good template pretty quickly and you can design your own projections, as well.