Does the size of a ECDSA key determine the hash algorithm? I am a bit lost in understanding what I read on authentication, signature, etc. For instance, is the size of the ECDSA keys produced by ssh-keygen -t ecdsa -b 256 linked to the hash used (SHA256) when computing the signature?
No, in general the hash isn't determined by the curve definition by NIST. Reasonable mappings of course exist (for a 224 bit curve you would probably use a hash with output size of 224 such as SHA-224). The hash used should however be specified by the protocol itself.



The ECDSA key size as indicated by the -b of the openssh argument is linked to the hash algorithm used. This is defined in section 6.2.1 of the RFC 5656:


  The Elliptic Curve Digital Signature Algorithm (ECDSA) is specified
     for use with the SSH ECC public key algorithm.
  
  The hashing algorithm defined by this family of method names is the
     SHA2 family of hashing algorithms [FIPS-180-3].  The algorithm from
     the SHA2 family that will be used is chosen based on the size of the
     named curve specified in the public key:


+----------------+----------------+
|   Curve Size   | Hash Algorithm |
+----------------+----------------+
|    b &lt;= 256    |     SHA-256    |
|                |                |
| 256 &lt; b &lt;= 384 |     SHA-384    |
|                |                |
|     384 &lt; b    |     SHA-512    |
+----------------+----------------+


Note that currently you can only use the P-256, P-384 and P-521 (secp256r1, secp384r1 and secp521r1) prime curves defined by NIST in SSH. So the argument for the bit size also determines the domain parameters used.