Is there a specifically "Chabad" way of learning Gemara? Is there a Chabad commentary on the Gemara?
There is a Chabad way of learning Gemara, as described by the Rebbe Rashab, the 5th Lubavitcher Rebbe and founder of Yeshivat Tomchei Temimim. In his Kuntres Eitz Chayim, he explains why it is necessary to learn Chassidus in addition to learning Gemara properly. The entire text is available online in English.
From the introduction to the english translation:
There is, however, a possibility for negative consequences even within Torah study. Since the Torah is enclothed in worldly affairs and operates with the framework of human logic, it is possible that a person will look at it as no more than a system of wisdom, forgetting about G-d, the Giver of the Torah. When the Torah is studied with such an approach, it can become “a potion of death,” encouraging a person’s self-concern. Instead of serving as a tool to bring about the refinement of the world and the person studying, the study of the Torah can inflate the person’s ego and cause him to become more materially oriented.
For this reason, it is necessary for one’s Torah study to include P’nimiyus HaTorah, the inner, mystic dimensions of the Torah, which focuses attention directly on the Torah’s G-dly and spiritual core.
The second half of the Sefer discuss the practical way Gemara should be learned. I will quote the summation of the chapters here just to give you an idea of what it advises, but the whole things needs to be learned in full, to understand the Derech the Rebbe Rashab is recommending.
[This chapter] emphasizes that the ultimate purpose of knowledge is to know the concepts [as they truly are], and not to try to develop chiddushim, innovative explanations.
[This chapter] explains the manner of studying the Gemara, [Rashi’s] commentary, and Tosafos.
[This chapter] outlines a course of study for a person with a developed intellectual potential. [It] reaches the conclusion that a careful analysis of the wording used is more appropriate while studying the Mishnah. While studying the Gemara, by contrast, one should [focus more on] concentrated analysis of the subject matter.
[The chapter emphasizes the importance of] being both “sharp and questioning, and patient and seeking resolutions.” [It also speaks of the importance of] carefully studying the wording used by Rashi.
[This chapter] communicates the [proper] approach to study — [continuing] the previous explanations — with regard to [studying the works of] the halachic authorities.
[This chapter emphasizes that a student] should protect himself from empty and false pilpulim. A genuine pilpul is possible only after one achieves an ordered comprehension [of the subject].
[This chapter] enjoins to remove [all traces of] the undesirable approach to study, and [states] that the study of Chassidus will assist in this.
Also, in one of the introductions to the Shulchan Aruch Harav, written by the son and successor of the Alter Rebbe, the Mittler Rebbe, the recommended ways to learn Halacha is also described, with different levels of learning described depending on the amount of time one has each day to devote to the learning of Halacha. I wrote up the most extensive way here.
Since no one seems to have answered the last question, namely, are there any Chabad commentaries on the Gemara....
There are many talks and writings from all the Lubavitcher Rebbeim on different topics in Gemara. As far as I know, there is no one systematic commentary on the Gemara, and as their discussions of Gemara topics are usually (though not always) in the context of their Chassidic teachings, they are usually published in their books on Chassidus and not by themselves.
When a certain Chabad family in Detroit wanted to send their son to Rav Bakst's yeshiva, Rav Bakst felt that because of the derech halimud (style of learning) of what the boy had experienced in a previous Chabad yeshiva, which was more about learning up the texts (girsadige) he would not do well in his regular misnagdishe-litvishe yeshiva. The family did not accept Rav Bakst's opinion so he wrote his opinion to the Rebbe asking for his support in handling the parents. The Rebbe responded "Misnagdishe Rav, Don't you know that the correct way to be learning is girsadige?! After all... (and the Rebbe quoted the Gemara in Shabbos about one who did not forget his "girsa diyankasa"(literally: text of his youth)."
This seems to imply that at least at the high school level the Rebbe held that the correct approach was a more textual and less analytical approach.
As a post-script, Rav Bakst responded by writing to the Rebbe that the "Girsa" (text) referred to in that gemara is a sevara (theory, used in deeper analysis of the Gemara). The Rebbe never responded.
Heard from Rav Aryeh Leib Bakst ZT"L
The Rogatchover is the quintessential Chabad Gaon (of the Kapuster variety). He learned with R' Chaim under the Bais haLevi and then with R' Y.L. Diskin. I also knew of a Chassidishe Rav (non-chabad) who was a magid shiur at the chabad yeshiva (in Brooklyn). So apparently there is not a specific chabad mesora in learning.
As far as I know, this isn't the "official" chabad shita, and I don't think there is one. However, what follows is the way that I learned in lubavitch yeshivos, and heard from others that this the way they learn in lubavitch. It is also the style in which tzemach tzedek al hashas, (3rd lubavitcher rebbe's pirush on shas) and likutai sichos, ( 7th lubavitcher rebbe's talks which include gemorah ) are written.
It is a sugyah styled learning, that involves comparing meforshim, and very often trying to find a certain shita that many meforshim ascribe to. The method is by going a (logical) step deeper. It is a special kind of pilpul and therefore they learn a lot of "Rogotchover" which is written in this style.
This is in regards to iyun. However the girsa doesnt seem to be very uniform, although it seems that chabad tries to focus on a simple and clear understanding with speed optimization.