Rails Console is so useful for direct sanity-checking of your model. Is there an ASP.NET MVC equivalent?
Is it possible to mimic Rails Console behaviour using LinqPA
Not really, since you're not inside the running application in the same way as with the Rails Console. - As Lloyd shows in his answer, it seems to be very much possible. Still seems that it's easier to use the Immediate Window if it is sufficient for what you are trying to achieve.
LINQPad is great anyway and I use it similarly to how I use the Ruby Interactive Ruby Shell (IRB).
The Immediate Window in the Visual Studio debugger can get you close to the same experience as the Rails Console gives. I hope that C# 5.0 will get us even closer because as of now you can't use lambda expressions and such.