Yes, please choose the redistribution license.
You can remove the "powered by" or text/logo by changing the Cute Chat/Messenger UI.
>>Also, can we customize the look and feel of the windows for our users so that it will fit in with our current website?
Yes.
asp.net Chat http://cutesoft.net/ASP.NET+Chat/default.aspx
Web Messenger: http://cutesoft.net/Web-Messenger/default.aspx
asp.net wysiwyg editor: http://cutesoft.net/ASP.NET+WYSIWYG+Editor/default.aspx
asp wysiwyg html editor: http://cutesoft.net/ASP
asp.net Image Gallery: http://cutesoft.net/ASP.NET+Image+Gallery/default.aspx
Live Support: http://cutesoft.net/live-support/default.aspx