For the sun, all are good except, 'it's a shiny day.' Shiny means something else.
Wind: all technically correct but I would generally stick with, 'it's windy'.
Clouds: I wouldn't say, 'The sky is covered with clouds,' or '
Clouds overcast the sky'. It is better to say 'it (or the sky) was overcast'.
Rainbows: I don't think I've ever been as expressive or poetic as you when talking about rainbows. The most I would say is 'Look, there's a rainbow!'. Oh and don't forget that there's a pot of gold at the end of the rainbow.
Thunder and lightning expressions seem good though I wouldn't use a peal of thunder. I'd say 'there was a crack/roar of thunder.'
Your fog expressions are perfect :-)