[SalesForce] what is the daily limit of time based workflow email alert

For a virtually unlimited number of email sends, you could relay the messages through your own service via call outs. Salesforce has to limit the messages sent to reduce the chance of spam filters blocking one customer's emails and consequently black listing many customers (because filters are usually by IP address).

Basically, this would simply require an endpoint hosted in a server you control to accept REST or SOAP messages, and in turn relay that to the local SMTP. You could also find a proper email hosting company that has a REST API.

I realize that the original question was about sending "through" salesforce, but from a practical standpoint, there's no alternative than to relay. It's easy to implement and gives far more flexibility.

In a sandbox the user's email address is changed as part of the refresh process (see guidance here) do that would be why your user cannot receive the email. Update the address to be correct and retry.