• The visitors released a biological agent into the air in the form of vapor clouds which would then manifest as the 'Red Rain'.
• Contrary to the visitors' claims that the red rain would correct all the world's ecological problems, it was in fact an agent designed to increase the phosphorus in humans, thus making them suitable candidates for breeding with.
• I don't think there is much in the way of a connection to the original series, other than the visitors' interest in developing a human/alien hybrid.
It could be a nod to the red dust in the original series, but this is inconclusive.
(Answer extrapolated from my own recollections plus this episode guide.)