Understanding that until I read this I never even knew there was a DB on the Mac called "Firebird." I am approaching this as a Windows support tech of a number of years and what this sounds like it might be. This will be a bit general and only point in a possible direction.
The first error message could be so many different things you could write a book. But the second error message may be pointing in a new direction: authentication.
With the Mac all you need to provide is a username and a password and you are in and depending on how you are authenticating you may not need to authenticate at all. Windows Does Not Work Like That.
Windows is all about authentication even if that authentication is (username) guest and (password) 1234. Oh wait did I mention that Windows expects the username to be in the form of "DOMAIN\Username." Domain could be the name of the user in an Active Directory domain or it could be just the name of the PC you are connecting to.
If you need to log into a domain-connected Windows PC computer as a "Local" user (EG not authenticated via an AD Domain) you need to enter the name of the computer (sometimes called a NetBIOS name) a backslash and the username. This is so Windows knows where to authenticate, locally or in the domain. This can be annoying especially if you don't immediately recall the NetBIOS name of the PC you need to log into.
So how would you log into that Mac from a Windows PC and tell the Windows PC to authenticate from the Mac's user list? Like this:
Where MACNAME is the name of the Mac listed in the sharing Preference pane at the top next to Computer Name. Note that keeping the name short and free of spaces will help you here.
And the password being the password of the user on that Mac.
So to connect to that Mac reliably over a network connection from that PC you would map the network drive and check the box to authenticate using different credentials. In the first box you would enter the computer name of the Mac, a backslash "\" and a username with the appropriate permissions on that Mac (also remember that spaces in computer names and passwords on Windows is verboten!.
And put the password in the password field. If you have entered the right info you should connect to the Mac and be able to Explore it's file system on the PC.
This is a long explanation to make sure you understand what is going on in the background when Windows authenticates. I believe (remembering my limited knowledge here...) that it is possible that you need to authenticate to the Firebird database in a similar fashion from the PC. IMHO.
Also you may want to contact the folks who support Firebird and see if I am totally full of little red ants or not and find out how you would authenticate in this fashion from a Windows PC.