Re #3, I don't think 'candle support' is at all idiomatic, at least in AE, unless it were to mean something else, like a buttress or guy wire that kept the top of a very big candle from tipping to one side. There are those little foam discs you can put inside a candlestick to give the base a sturdier anchor and cut down on wobbling or tipping; I don't know what I would call those, but I don't think 'support' would be it, as that would sound more like moral or emotional support.
But yes, any or all of the other terms would be okay and are indeed all used. To me 'candlestick' and 'candlestick holder' are interchangeable and have the same meaning. You got us. \-;
Like dude, I thought about 'stick' as possibly alternatively meaning any long, slender, stick-shaped type of candle, and I also thought about 'stick' as possibly alternatively meaning the spike that some candlesticks have in the base to stick the candle onto. I would guess that either of those is indeed etymologically possible, but I'm afraid the easiest answer is probably more likely: namely, that it's just one of those illogical repetitions that has become common and widely accepted at least in colloquial usage, like 'tuna fish' or 'ATM machine.'
I would write candle holder as two words, and I would use it mainly only for something that is not actually the shape of a candlestick. For example, a glass or pottery dish, bowl, or vase-shaped or box-shaped container that you put a small candle such as a tealight on top of, or completely inside of.
Or maybe also one of those low, flat, circular, wood or stone discs that you set a fat pillar candle on top of -- though we might more likely call that a candle base.
I'm sorry if it's disappointingly illogical, but if it moved Cali81 to pay another visit to the forum, maybe that's a good thing. Good anyway to see you around.
And if you find this one frustrating, do be sure to inquire about 'candelabra' and 'candelabrum.' (-;