Harmony in general is a pretty broad topic and there isn't just one option for how to do harmony. In general, harmony is the simultaneous or "vertical" relation between what is being played. There is the typical Western idea of functional harmony where the Tonic-Dominant relationship (I-V) drives the progressions we encounter, but there is a lot more out there.
Does harmony science help me solve these 2 problems above?
In general as a composer you have to choose how you want your harmony and melody to sound. It is not a science. There is no right or wrong, but you should be aware of what you are doing. For example, with just a G I can harmonize almost anything with it. It doesn't have just C, Em or G especially since I may not want to work in C major or G major. I may be tempted to play something like an F9 where the G would end up being the 9th. It all depends on what I want my melody and harmony to do.
There is one thing that should be pointed out. Whatever your harmony and melody is the should be related somehow. How will be up to you as a composer. For example when writing background parts when someone starts on a C note there are plenty of options. I could give two background vocals an E and G, or an E♭ and G, or an A and E, or A♭ and E♭, or an F and A, or an an F and A♭. All of those and many other work, however the harmony is much different and it may or may not be what you want as a composer. You need to know what you want then you can craft your melody and harmony the way that works for your ideas.