Dhouhitran is more commonly spelled as Douhitra (दौहित्र), and it just means the son of one's daughter. (Source)
Naazhigai is a Tamil word for 'twenty-four minutes (Source). It is one unit in a more complex time-counting system. I can't find any uses of it being used in a ceremonial context, but I assume that it has something to do with how long one has to do a certain action.
Pumsavana is a ceremony, part of the Shodasha Karma, that is done during a woman's pregnancy, traditionally to secure the birth of a male child. It is typically done when the mother's stomach is starting to become larger and the baby is beginning to show (Source).
Kanganadharanam is, I suspect, a variant of Kankana Dharanam. This is a part of the marriage ceremony, during which the bride and groom tie threads of saffron around each other's wrists. (Source)
Swasthi, or 'svasthi,' doesn't have much specific to birth ceremonies, as far as I can tell. But, it has several meanings in Sanskrit, including well-being, success, prosperity, luck, happiness, fortune, and welfare. (Source) 'Sri' was likely added on.