Yes, a quantum computer with a few hundred error-corrected qubits (many more before error correction) would clean out all addresses with exposed public keys, by solving Elliptic Curve Discrete Log to recover the private key.

This includes many old addresses that didn't hash the public key, any re-used addresses, and, using replace by fee, even all new addresses in the time frame between broadcasting a spend transaction and that transaction being included in a block (which can currently take many hours).

This is an interesting question to me, because I know little to nothing about the cryptography involved with cryptocurrencies.

1) What's the probability of the cryptographic ciphers behind cryptocurrencies being broken?

2) If they were broken, is it possible to 'move' BTC or other coins to a new cipher?

I apologize if I'm using the 'cipher' terminology incorrectly.

As far as I’m aware, hash algos like SHA or Scrypt are not the type of math that quantum computing threatens.