Would you be down to X = would you be okay with doing X
Would you be down to have pizza for dinner? =
Would you be okay with having pizza for dinner?
It is very casual and 'cool' English...
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Saving yourself for marriage = not having sex until after marriage
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In the case of "Would you be down to like pinky promise to save ourselves for marriage," "like" is just an unnecessary word some people add when speaking.
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Basically, "Would you be down to like pinky promise to save ourselves for marriage?" = Would you be okay with promising that we will not have sex until marriage?"