"Costume" typically means the appropriate clothing - and sometimes hair and makeup - for competition at the open level, for example tails and ballgown for Standard. Costume is often forbidden in lower level competition. "Competition dress" means whatever one competes in, which might be a ballgown at the open level or more casual dress at a lower level.
For a show, appropriate dress would depend on the specific show. If you're doing a studio show, I'd recommend relying on the instructor's advice - or the advice of his partner - as they'll know what's expected. In general, I think it's safe to rely on the instructor's advice until you start forming opinions of your own, and definitely bring the dress in beforehand so the instructor can check to make sure it doesn't interfere with the dancing.
When I was competing in Standard and Smooth, my partner always wore the same ball gown in both, although some ladies prefer to have two different gowns. There are some standard gowns that don't work for open level Smooth because, while they are fine in a closed dance position, they cause problems in certain arm actions, such as bits falling across the lady's face. In principle, Standard only requires one full circle skirt while Smooth may require a double full circle to accomodate actions where the lady takes hold of her dress, but I've never actually seen a Standard gown that isn't full enough for Smooth.
Shoes are very much an individual or partnership thing. In addition to the woman's unique foot and body shape, some women may adjust their heel height based on the height difference with their regular partner. The nude colors popular these days are used to deemphasize the shoes, and perhaps the feet, which makes the specific design less important for judging. Women who have good feet may match the color of their shoes to their dress color instead. I can imagine that some judges might mark down for using cuban or block heels - the wider ones - but subtler differences are less likely to make a difference to judges. Black shoes are great for practice but can look heavy in competition.
I've always just assumed it was because many women like to change dresses a lot, rather than need. That said, I've had partners that wore the same dress for years.
One thing to note is that it's not a lot more expensive to change dresses frequently than to keep the same dress, because most women wear used dresses, so you can usually sell the dress for not much less than you bought it for. This facilitates having "new" competition dresses frequently.
Occasionally, perhaps. I remember one competition when my then partner was wearing a dress that had previously been worn by a world champion several years and owners before. That world champion's partner, now retired, was judging the competition, and did a double take clearly visible on our video when he recognized the dress. I don't think it matters, though.
"Salsa" seems to be a currently popular social dance, so it can be the dance most useful to many students. And yes, it's the same as Mambo except that some people dance Salsa breaking on 1 instead of breaking on 2.
Bolero and Viennese Waltz are not generally danced socially in American style, so they are arguably the least useful dances in Rhythm and Smooth respectively. They are also typically not included in competition until the higher levels. It's not surprising that you haven't worked on them 6 months in; some people never learn them. Mambo is the next least popular dance, and yes, Salsa is basically the same dance, if you learn to break on 2.