Are there performance, size, or power efficiencies from one curve to another?
The larger the curve, the larger the keys and signatures, and likely the slower the computations. There are exceptions to the last one – curve parameters do affect how efficiently they can be implemented, so some curves with good parameters can be faster than slightly smaller curves.
Perhaps more important, are there security improvements over a given curve?
All of them should be safe, but the larger curves have more security margin.
If a new alt-coin (Bitcoin) were created today, which curve is ideal?
Since signatures and public keys appear in the blockchain, having them be small would be great. Having signature verifications take little time would also be good, since that's time wasted.
Curve25519 has 256-bit keys and 512-bit signatures, and is known to have a fast implementation (Ed25519 for signatures). Especially if it gets standardized, it would be a decent choice.
Unless there's some kind of a breakthrough, 256 bit curves with 128-bit security should be enough until/unless quantum computers break ECC. For a post-quantum altcoin, you could use e.g. hash signatures, but that would expand the blockchain quite a bit.