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: 
Every guitar player is different so it depends. I have a friend who took piano lessons and read music before learning guitar and she would much rather know the names of the notes. 

I play based on sound and feel and knowing where to put my fingers relative to where they are now to get to the note I want next relative to the note I am on. 

But I have a musical ear. If you want me to play something, I would rather hear it than see it on tab. If you give me standard notation and ask me to play what is written I will get back to you in a few days. If I hear it, I can play it back immediately. 

I personally have no need to know the names of the notes I am playing while I am playing them (I **could** tell you if I **thought** about it). I just need to know how to make what I hear in my brain translate to my instrument and the easiest way for me to do that is to start on the right note and go from there. 

I am quite certain that if I was improvising and I tried to **engage the part of my brain that would allow me to identify the notes** I was playing, it would stifle my creativity and instinct. I think the best improvisers play by **allowing their mind to engage with their soul and play what "feels" right. Thinking about the NAME of the note they wanted to play next - would short circuit that connection.** Who cares what the note is called as long as it sounds good!

If you can improvise without knowing the **names** of the notes you are playing, your time will be better spent improving your craft by playing more - than by **trying to learn to identify each note you are playing while improvising.** That is my assessment based on my own personal playing style. Your mileage may vary.
I'd memorize the relation of the roots to the tonic for analysis and overview. However, I find it more musical, if I have to try and verbalize, to remember the relation of a note and the current (and sometimes previous and/or next) root.