What I think, my thoughts, are they coming from the immediate present?

Nothing at all happens anywhere but the immediate present. It is even more intimate than that actually. Your "memories" come from there too, in the sense that they reside always in the now. There is only this moment and this is not even a religious idea. What you believe and how you see yourself are only 'now' as well. There is no black box somewhere, separate from the you who asks, that contains all this info, separate from this moment.

You are almost, but not quite, asking "Why" this is so. It is surely a good question in case you choose to follow it. It may be the best question of all. The way in is to ask: "Who am I". If you can adopt the standpoint revealed by the authentic pursuit of this one question, then memory, belief and time will answer themselves as only your awareness of such concepts and nothing more. Then, you will be free.

You can have thoughts that originate in the immediate present. I think they are from the past, from what has become already a part of our experience, our beliefs, and yes they affect the way the present is viewed and how I see myself. There are thoughts I haven't thought of yet! Sometimes I have thoughts that originate in the hear and now based on a belief, and then I expand on the belief and its own possibilities.

Human mind surely links the present with past through remembering and the future through predicting.

If we meet a person, without our intention we think about the past incidents that happened with that person, and we also predict him only based upon his past behavior. Through this surely one can understand that we are all not living only in present but also live in a prediction world.

You asked "Does this affect the way I see the present?", surely it does. You see yourself as a person who departed from all other existence, but you are a person who is a part of this existence. Surely every one fails to think them self as a part of this existence. The real present is some what different from what we think

To understand this, try the following:

Your mind always thinks something, so stop that and keep it empty for at least 2 min. Keenly watch the things going around you. Surely you can recognize the reality. You can yourself clearly find the answer of your question.