do people actually say "ta"?

[Einstein]

I confess that I haven't read through all this thread! Just a quick comment about ta-ta, meaning goodbye. I do find this a bit childish and I see it as being used half-jokingly between adults or when speaking to little children. I don't know why the Americans think it sounds like upper-class British - the upper classes would be the last to use this expression.
As for ta-ra, this comes from a northern tendency - especially Liverpool - to substitute t's with r's, so "what about..." becomes "worrabout...".

Ta, for thank you, is definitely not childish. I too have heard the Scandinavian explanation, but rather than coming from Viking invasions it came from Swedish sailors calling in at Newcastle. I've no idea whether this is more accurate, it's just another version to think about.

 

[Odinn]

That is not logical. If "Ta" were lacking in Bantu would that enhance anything?

If the assumption that "Ta" comes from Scottish Gaelic is true, it would not be found in Irish Gaelic. If it was found in Irish, it would not be specifically Scottish, now would it? It would be Irish. That "Ta" is lacking in Irish enhances the argument that it comes from Scottish. You don't see that/agree?

OK, where is it shown that "Tapadh leat" in Scottish Gaelic is contracted to "Ta"?

Nowhere that I'm aware of.

I've had a look at a Scottish Gaelic dictionary, and there is no note of this, but tà,thà appears as the present tense of the verb to be, which, assuming a similar pronunciation to our "ta" would lead to confusion...

Possibly. As Wordsmyth also pointed out, and as I also mentions earlier, "Ta" is to short to find a specific root. It fits in too many places and could for that matter be absolutely coincidental.