Twisted Edwards curve and stealth addresses

Question

Monero uses Twisted Edwards curve Ed25519. My questions:

1. Is it necessary to use this curve for stealth addresses or can one use any kind of elliptic curve, such as secp256k1?
2. How to define Ed25519 and other Twisted Edwards curves with Sage?
3. In the Monero source is a predefined constant ge_precomp ge_base[32][8]. How can I compute it?
4. Is it possible to define small toy Twisted Edwards curves with 16 or 24 bits instead of 255 bits, with another parameter d which is equal 121665/121666 for Ed25519?

Answer 1

1. Stealth addresses require an agreed upon eliptic curve and hash function. For use with Monero, this is Ed25519 and Keccak (as used in Hs - hash to scalar), respectively.
2. I have no knowledge of "Sage". Perhaps someone else can answer this for you.
3. See https://crypto.stackexchange.com/questions/53154/how-is-the-precomputed-table-for-25519-elliptic-curve-generated
4. Possible, yes. Useful, doubtful.