I am looking for the antithesis of G-d in Judaism (if there even is one). Also please elaborate the role of Ha-Šatan in the grand scheme of things. And moreover, I am very curious as to what theological weight does the notion of 'evil' and 'the devil' bear in Judaism. For example, Satan in Christianity is pretty straight-forward, however I do not know if this is the case in Judaism. I've read all of the Scriptures but still feel like I need some guidance from someone who practices Judaism and knows it in more detail.
One of the fundamental results of belief that G-d is one (like is said when reciting the Shema twice a day,
Hear Israel, the L-rd is our G-d, the L-rd is one.
is that there is no such thing as the antithesis of G-d. There is no 'other' apart from Him like it says in Devarim 4:35. (אין עוד מלבדו)
The Satan, the name describes literally what the function is in the grand scheme of things. שטנה means to accuse. The Satan serves to make accusations.
When trying to understand the purpose behind that function, a good example is learned from the story of the binding of Yitzchok. Like was seen in parshat Vayera last week, the angel which spoke to Avraham telling him to bring Yishmael, Eliezer and Yitzchok and to offer Yitzchok as an offering to G-d at Mount Moriah was the Satan.
The commentaries on this parsha explain that the Satan made an accusation against Avraham as a consequence of the peace agreement which Avraham made with Avimelech. It revolved around the question of Avraham's faith and trust that G-d would fulfill His promise to give all the land of Israel.
Like with all the trials of Avraham, one aspect of each trial was to bring some trait of Avraham from a state of potential into a state of actualization and realization. In other words, for example, not just the potential character trait of being kind, but to actually demonstrate kindness. This idea of the importance of actualization is based on the general principle that Action is the primary thing. (המעשה הוא העיקר) Avot 1:17. And this is the explicit statement from G-d to Avraham (Bereshit 22:16) following the binding of Yitzchok and the offering of the ram in substitute.
As a consequence that you did this thing...
And this idea of the function and purpose of accusation being connected to the Satan (in your words, the embodiment of evil) can also be understood from the related phrase to challenge (לְעַרְעֵר עַל) which in Hebrew shares the letter root for evil (רע).
The Hebrew word SATAN can also mean "opposition"; that is, like your opponent in a tennis match. Only with the right opponent can you achieve your fullest potential on the court. So too the game of life, God gives us these challenges and temptations in order to help us build our spiritual muscles - according to the effort is the reward. We call those moments of challenge or resistance "the Satan". It is an expression of God's awareness of a person's area of spiritual weakness that needs work.