I can't answer your questions with certainty, but may be able to throw some light on the subject.

The Indian curry leaf is definitely NOT a Helichrysum. I am reasonably certain it is always Murraya koenigii.

Meetha/mitha neem is one of the many Indian names for Murraya koenigii - see: http://www.hort.purdue.edu/newcrop/parmar/12.html

'Sweet neem' is a name sometimes used for Murraya koenigii.

But a Google search indicates that 'Sweet neem' is also used for Azadirachta indica var. siamensis (= A. siamensis), an unrelated, larger tree though with superficially similar foliage and flower-buds. In Thailand you often see bunches of the tender young leaves and buds sold in vegetable markets and I gather it is popular as a medicinal herb, though the taste is fairly bitter. Its Thai name is sa dao. I don't know about its uses or names in other Asian cultures.

An additional note: recent botanical evidence shows that M. koenigii is not closely related to the other Murrayas, typified by the well known ornamental M. paniculata -- so it was concluded that it should belong in the separate genus Bergera, as B. koenigii.

TonyfromOz had answered correctly.