As of 1.69.0, you can now update the password hash using the User API.

From the documentation, the passwordFieldType is the API field to use. This field:

Describes what user.password in the request is. The possible values are:

HASHED - user.password is an already hashed value. When this value is supplied, user.encryptionScheme, user.factor, user.password, and user.salt are required.
PLAINTEXT - user.password is a plain text/unhashed value. FusionAuth will hash this value according to user.encryptionScheme and user.factor.

You'll need to separate out the hash and the salt on the Import API.

For example, the hash $2a$10$N9qo8uLOickgx2ZMRZoMyeIjZAgcfl7p92ldGxad68LJZdL17lhWy would be split out to the fields:

factor: 10
salt: N9qo8uLOickgx2ZMRZoMye
password: IjZAgcfl7p92ldGxad68LJZdL17lhWy

The Import API does not parse this value and separate it out for you. You need to do so.

Here is an example import script (in Ruby): https://github.com/FusionAuth/fusionauth-import-scripts/blob/master/auth0/import.rb#L47

Yes you can. You have a few options.

If your algorithm matches one of the FusionAuth supported algorithms or one of the community provided ones, you can configure FusionAuth to use it. If not, you may need to write your own hashing plugin.

Once you have imported the hashes, the next the to consider is whether you want FusionAuth to rehash the passwords as the users log in. This is a good option if the previous hashing algorithm was not a strong one. This is configured at the tenant level, under the Passwords tab.

Currently the only way to accomplish this will be to use the Import API, as you mentioned: https://fusionauth.io/docs/v1/tech/apis/users#import-users

We do have an open feature request to allow hashes to be provided on the User API, which I think would be what you're looking for: https://github.com/FusionAuth/fusionauth-issues/issues/348

Feel free to upvote that issue.