if we can use TFS or Team Explorer with VS2010 Professional?
Yes. There are three levels of client support for different versions of Visual Studio and Team Explorer. You can refer to this document about Azure DevOps client compatibility. Please also refer to this ticket about Visual Studio 2010 Team Explorer.
Is Team Explorer 2010 free?
Yes, you can download Team Explorer 2010 for free. But users of Team Foundation Server will typically need a client access license (CAL). See the licensing white paper (Pages 25 and 28) for details. You can also refer to this blog.
By the way, Visual Studio 2010 has ended its lifecycle. See this document for details. So if possible, I may suggest you consider using the latest version of VS, for example, VS 2019. It is stable and powerful, of course, this may cause some compatible issues, and just as a suggestion.