Can I run my old Visual Studio projects and source codes in Mono correctly?
For the most part. There's some magic that you can do the mono compiler won't handle. Most people don't do that though. MonoDevelop will open up VS solution and project files normally without a hitch.
My clients use IIS for ASP.NET and MS Windows for Winform and Console projects. Does my migration affect them? In other words does my Mono projects runs in Windows/IIS without any change or problem?
You will want to test everything before deploying to production, but normally this isn't a case. Any problems will be edge cases.
Does Mono is dependable considering recent changes of Mono, Novell, Xamarin and Attachme?
Xamarin has aquired the rights and licenses to everything mono related from Attachmate. This itself is not an issue. Xamarin is putting resources forward to make MonoDevelop a better product. I spend the past weekend with many members of the mono team, and they are looking to continue the product line; however, a stronger focus will be with the mobile things. Server side stuff won't be ignored but I wouldn't expect the fastest adoption of new stuff.
Can I continue using my favorite open source and none open source projects like NHibernate, Castle, Cuyahoga, log4net, Microsoft Enterprise Library and DNN?
I cannot comment on each project individually but many projects will run on mono with little to no modification.
Can I host Mono ASP.NET web projects in Apache without any problem?
It depends on your apache skills but it's possible to do.
Can I use my skills in ASP.NET webforms and MVC in Mono exactly?
The should be no difference in the code you write between doing so in VS and MonoDevelop.