Is a Windows 7 homegroup(workgroup) and a Windows Vista Domain compatible?

Can a Windows 7 computer connect to a Windows Vista domain?

Can a Windows Vista computer connect to a Windows 7 homegroup(workgroup)?

There is no such thing as a Windows Vista domain. There are NT and Active Directory domains of different vintages, but they all run via Windows Server. None of them run on Vista. Windows 7 can connect to any of them.

Windows Vista can connect to a "Windows 7" Workgroup. I use quotation marks because workgroups here are not specific to the version of windows or even Windows itself. I think it's likely you mean "Homegroup". Homegroup is a Windows 7-only feature, but it is possible to access some Homegroup resources in Vista.

Please remember that when dealing with technical concepts, accurate terminology is important.