Thank you for your patience while we researched the issues that you bring up. Of course, the issue is complex and for any practical guidance a competent Halachic authority should be consulted. We will offer a general overview.

The source of this Halachah is in the Mishnah in Pirkei Avos (1:8), "Al Ta'as Atzmechah k'Orchei ha'Dayanim." The Rishonim offer many different Perushim on this Mishnah. There are those who write that this Mishnah, like the rest of Maseches Avos, is merely a suggestion of Midas Chasidus (pious behaviour) and is not Halachah. However, others counter that it is clear from the Gemara which you quote that it is indeed Halachah. Among those who say that it is Halachah there are different explanations of the exact meaning of "Orchei ha'Dayanim" and to whom this Halachah applies.

The Shulchan Aruch (Choshen Mishpat 17:9) brings the Halachah with regard to a Dayan helping one of the litigants. The Pischei Teshuvah cites a difference of opinion if the Halachah mentioned in our Gemara relates to a non-Chashuv person as well or not.

The basic rule in the Poskim is to differentiate between using legal loopholes and tactics in contrast to merely helping to clarify the claim. The former might be Asur, even if it is clear that the claim is justified, while the latter is generally accepted to be Mutar.

As for representing a client in Beis Din when the client is the plaintiff, the Gemara (Bava Kama 70a) allows it when receiving a contingent fee. The Urim v'Tumim writes that although there is no clear Talmudic precedent for representing the defendant, it is accepted Minhag to allow it.

The question of non-Jewish law depends on the many explanations given for the Isur. Many Poskim write that the "Orech Din" (lawyer) may sway the Dayan and therefore cause the Din to be swayed. This reason would not apply to a non-Jewish court (in the event that Halachah allows going before it), for its decision is not Halachicly binding due to its justice (i.e. we do not consider any alternative legal system to be entirely just), but rather due to the agreement of the parties to abide by the court's decision. It therefore would be permissible to act as a lawyer in such a case. In the event that one of the litigants is non-Jewish, or refused to follow Torah law, the prohibition would surely not apply (however, where both the claim and the tactic are unjust, one may run into a problem of "mi'Dvar Sheker Tirchak"). (Much of this is based on the Sefer "Bnei Binyamin," by Rav Binyamin Lifkin, a grandson of Rav Yisrael Salanter who lived about 50 years ago.)

Once again the above is very general, specific questions should be decided by a competent Rav.

Much success in all your endeavors

Dov Zupnik