Are our dominant limbs decided on birth or is there some way in which I can train my non-dominant hand and make it as coordinated as my dominant?
Of course you can train yourself and make your non-dominant hand to be equal or even greater than your actual dominant handedness. Handedness is directly wired to the right/left hemisphere of your brain, so by training, you will actually re-wire your brain and acquire new pathways. It's a hard process in terms of breaking habits, otherwise, it's non-invasive and non-painfull, where most humans tend to give up before acquiring this skill justifying it as an unnecessary/low priority goal.
Handedness is not decided upon birth. It's gained in the earlier process (while a child is still in mother's womb) based on personal/individual preference (expressed by sucking a finger btw) toward right-handedness (90%) or left-handedness (9%), cross-dominance/ambidexterity (1%).