The integration of blockchain technology into healthcare presents a paradigm shift for secure data management, enabling decentralized and tamper-proof storage and sharing of sensitive Electronic Health Records (EHRs). However, existing blockchain-based healthcare systems, while providing robust access control, commonly overlook the high latency in user-side re-computation of hashes for integrity verification of large multimedia data, impairing their practicality, especially in time-sensitive clinical scenarios. In this paper, we propose FAITH, an innovative scheme for \underline{F}ast \underline{A}uthenticated and \underline{I}nteroperable mul\underline{T}imedia \underline{H}ealthcare data storage and sharing over hybrid-storage blockchains. Rather than user-side hash re-computations, FAITH lets an off-chain storage provider generate verifiable proofs using recursive Zero-Knowledge Proofs (ZKPs), while the user only needs to perform lightweight verification. For flexible access authorization, we leverage Proxy Re-Encryption (PRE) and enable the provider to conduct ciphertext re-encryption, in which the re-encryption correctness can be verified via ZKPs against the malicious provider. All metadata and proofs are recorded on-chain for public verification. We provide a comprehensive analysis of FAITH's security regarding data privacy and integrity. We implemented a prototype of FAITH, and extensive experiments demonstrated its practicality for time-critical healthcare applications, dramatically reducing user-side verification latency by up to $98\%$, bringing it from $4$ s down to around $70$ ms for a $5$ GB encrypted file.