Are the Windows and Linux versions save-game compatible? Are my Windows saved games playable on the Linux version, and vice versa?
The Intel version of Linux Civ:CTP 1.1 is save-game compatible with the Intel version of Civ:CTP. Other architectures will not be save-game compatible. This goes for scenarios as well.

Can I play multiplayer under Linux against users playing Civ:CTP for Windows?
The original Linux and Windows versions of Civilization (version 1.0) are not network compatible. All versions beginning with the first patch (1.1 and greater) are network compatible, provided all network players are using the same version of the game. For example, Linux Civ:CTP 1.1 is network compatible with Windows Civ:CTP 1.1.

Would it be technically possible to use the Windows Civ:CTP CD and then download the Linux binaries?
No, the games use slightly different data file layout, and have some data file differences.