Question: When playing guitar,especially improvising,is it best to think of the notes a actual lettered notes or Scale degrees related to a tonic ? Since the keys are easily movable , I'm thinking that its better just to know the degree you want to play and knowing where that is related to the tonic.

Answer: 
Guitar is a three-dimensional instrument, where you are almost forced to know things in multiple locations.
Here's something I think is of paramount importance: when you learn a tune on guitar, you should always practice playing chord and melody 

together. That way you really learn to link the two things together. 
It has to become almost automatic. Then when you improvise, you should be able to see (and play) chords at almost any time as you go by.
About note spelling when you play, I think it has to do with different areas of the brain. 
It's the same thing that happens when you speak another language. These are two different processes: one process is knowing and using that 

language directly, without translating it. The other process is translating what you say into your own native language.
That's what simultaneous interprets do. But they never do the two things together; always sequentially.
So when you play, you bypass the names and go to the heart of things. If you really know these things, you won't need anything else. 
If you listened to music for enough time, you would tell most chords by ear, as if I’m listening to a D chord on the radio I know it’s D. 
And if I can hear their quality, find them on the fretboard, and play them, what else do I need?
I do think some of the answers so far are awesome. Let me please add by indicating something I was taught in music school.

The mode of a sequence of notes is also important, as the mode one is in can actually alter what the ear "hears."

For example the relative minor of CM is Am. The "notes" are exactly the same when no accidentals are present, yet we hear them differently when we look at the "root" note.

As I improvise, I often use modal shifts as a means of running from one key to another, or one section of the music to another (like at the end of a bridge). It's amazing what you can do by shifting run from (as above) CM to Am to AM when one rests on A as the "root" note for a moment before heading off into a key signature with 3 sharps.

[The Circle of Fifths](http://en.wikipedia.org/wiki/Circle_of_fifths)
[The Modes of Music](http://en.wikipedia.org/wiki/Mode_(music))