You need to handle authentication in your Spring Boot backend not in the react frontend (see: https://spring.io/projects/spring-security-saml) because SAML is only an authentication mechanism, it does not provide authorization like OpenId (there's no access token).
When the user is authorized, you could give back a session cookie to your react app that will provide the necessary information to authorize future API requests to your backend.
As for your second question, you could find a way to use the SAML assertion you used to connect to application A to retrieve an access token for Application B.
There is a specification for this kind of mechanism: https://tools.ietf.org/html/draft-ietf-oauth-saml2-bearer-20
Bear in mind that this would imply a lot of custom implementation work, you would be better to use OpenId across all your apps in order to have SSO for your users.
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