Is there a consensus on whether the serpent in the Garden of Eden was Satan or not? Is it possible that the application of the word serpent is an adjective?
First let me briefly address your use of "Satan". Another religion has an idea of "Satan" as being an evil godlike being or angel, one who is locked in a struggle with God between good and evil. That idea is not really part of Judaism. There is a role of "the satan", which might not always be filled by the same entity, but this is a servant of God, just like any of the other divine messengers we see throughout the Tanakh (and midrash(.
Now, about the serpent, the plain meaning of the text, both at the beginning of B'reishit 3 and later in the chapter when it receives its punishment, is that it is an animal. Sure it's a talking animal, but we don't actually know if that was unusual in the garden. But there is nothing in the text (the pshat, or plain reading) to suggest that it's anything other than one of the animals created that week.
I checked several chumashim and found only one reference to the satan. According to the Soncino Chumash (ed. A. Cohen), Sforno says that the serpent is evocative of the satan. He doesn't (according to this tertiary source) say that it actually is; he just draws the comparison. I have not chased the reference.
Rashi understands the serpent to have certain human traits (jealousy, in particular), but says nothing about it being anything other than an animal. Chumashim I checked included other comments from Ibn Ezra and the Ramban but neither raised any connection to the satan. (I realize that argument from omission is weak.)
I haven't addressed the last sentence of your question because I don't understand what it means.
If I may post an answer to a similar question, there are different approaches within Jewish tradition as to the identity of the biblical serpent. (The first three here are quoted by the Ibn Ezra)
It could be that the snake in the story is not literally a snake, but a metaphor for the Satan, or some similar evil force. This is the approach taken by r. Ovadia Seforno, that the serpent is the evil inclination. According the Gemara Bava Basra (16a), the evil inclination (יצר הרע) is equivalent to Satan, and several midrashim see to take this approach as well. However, this approach is difficult to read into the actual verses, as the Ibn Ezra himself notes, and he rejects this opinion. The Seforno's position is discussed here
Chava was a wise enough person to understand snake language (a parselmouth, בלע"ז)
R. Saadia Gaon says that an angel spoke for the snake, even though the 'evil plan' or whatever it was that he did wrong was his own fault (the angel just helped give him a voice)
Ibn Ezra himself does believe that pre-tree snakes could talk, and though he doesn't mention this lack of punishment, he might reply that by losing his ability to eat normal food and walk on 'legs' are symbolic of becoming less human-like and indicate a loss of the ability to speak as well.
Abarbanel writes that the snake never actually spoke, and though the Torah states that he spoke, it means that he gestured in way that his message to Chava was obvious (such as by continually eating from the tree, and shoving Chava into it, etc.)