Origin of 天气 (weather)?
气 is vapor, and so 天气 is the vapor of heaven, that is weather. 气候 is climate by analogy, referring to a timespan, or 地气, referring to the vapors of earth.

But 气 is also breath, attitude, atmosphere and similar derived meanings. Poetically, 天气 could mean the influence of the heavens, although it is likely that 天气 was applied to such settings rather than being derived from religion.

樂記 (战国):

地氣上齊，天氣下降，陰陽相摩，天地相蕩，鼓之以雷霆，奮之以風雨，動之以四時，暖之以日月，而百化興焉。如此則樂者天地之和也。

The breath (or influence) of earth ascends on high, and that of heaven descends below. These in their repressive and expansive powers come into mutual contact, and heaven and earth act on each other. (The susceptibilities of nature) are roused by the thunder, excited by the wind and rain, moved by the four seasons, and warmed by the sun and moon; and all the processes of change and growth vigorously proceed. Thus it was that music was framed to indicate the harmonious action of heaven and earth.