Should the secret key of Shamir's secret-sharing algorithm be interpreted byte by byte? Should the secret message of Shamir's secret-sharing algorithm be interpreted and processed byte by byte? 
Interpreting it byte by byte makes it easier to process, but in case one of the shareholders decides to tamper with some of the bytes in his possession, this will affect the secret reconstruction.
This is a serious risk, and I don't know how it could be prevented.

Would it be sensible, as an alternative, to interpret the secret message as a whole string, instead of byte by byte, transforming it into an integer? In this case, what I am worried about is the huge size of the integer...
Shamir's secret sharing scheme provides only confidentiality against shareholders who want to try to learn the secret.  It does not prevent denial-of-service attacks (or attacks on integrity), where a malicious shareholder submits a bogus share to try to cause the reconstruction of the shares to fail.

If you want security against that sort of attack, don't use Shamir's secret-sharing scheme; there are other schemes that can accomplish that.  Verifiable secret sharing schemes do exactly that: they protect against malicious shareholders who submit bogus shares.  There are many constructions of verifiable secret sharing schemes in the literature.  They do exactly what you want.

Depending upon your application, you may also find threshold cryptography  relevant.