おかげさまで

Yes, but it might sound a bit formal. With family, you'd more likely use 'Okage-de' or just say 'Arigatou.'

It has religious roots, but today it is a secular social phrase used by everyone, regardless of faith.

Then do not use 'Okagesama-de.' Instead, say 'Zannen nagara...' (Unfortunately...) or 'Chotto...' (A bit [bad]...).

Yes! If you were planning a picnic and it's sunny, you can say 'Okagesama-de hare-mashita.'

Not rude, but less formal. Use it with friends or when talking about inanimate objects like 'the internet' or 'the map.'

It is usually written in Hiragana (おかげさまで), but the Kanji is お陰様で. Hiragana is more common in friendly letters.

Sort of, but it's more like 'I am in this good position because of factors including you.'

Yes, if they ask how you are (though strangers rarely ask that in Japan unless they are shopkeepers).

The phrase itself doesn't change, but the following verb does: 'Okagesama-de yokatta desu' (Fortunately, it was good).

Because a shadow provides protection from the sun, just as a benefactor provides protection from life's hardships.