1) It is correct to bless the moon the evening after Shabbos when a person is still dressed in nice clothes and smells nice. However, if the evening after Shabbos comes out to be after the tenth day of the month, one should worry that there may be many cloudy days and if he waits that he will be unable to make the blessing, and should therefore make the blessing at the first opportunity. (Shulchan Aruch 426) The opinion of the Gr”a and other latter halachic authorities are that one should not wait until the eve after Shabbos but should make the blessing at the first possible opportunity.
2) In the ideal circumstances one should not bless the moon when standing under a roof, as blessing the moon is considered to be “Receiving the manifestation of the Divine Presence”. We therefore exit our houses into the street as one who goes out to greet a king. However, if for some reason it is difficult for him to go out, he may make the blessing inside the house when looking at the moon through a door or window. (Shulchan Aruch and Mishne Berurah ibid)