The DR number, Diederich Rank, refers to the highest number in Diederich's frequency listing, which you can see here: Diederich Ranking.
DR 176. Audi et alteram partem. ~ Note: Compare the saying you saw earlier: Audiatur et altera pars.
DR 176. Alter ego est amicus. ~ Note: For the various uses of the phrase "alter ego," see this Wikipedia article.
DR 176. Amicus alter ipse. ~ Note: This is a saying collected by Erasmus in his Adages, 1.1.2. Notice how ipse is being used here to mean something like the English word "self" - "A friend is another self."
DR 176. Quod tibi vis fieri, hoc fac alteri. ~ Note: Notice how fieri serves as the passive of facere: That which you want done to you, do to another. This is sometimes called "The Golden Rule," which you can read about at Wikipedia.
DR 176. Quod tibi non vis, alteri ne facias. ~ Note: This is the negative version of the Golden Rule, sometimes called the Silver Rule: What you don't want for yourself, don't do to another! You can also see this with the perfect instead of the present subjunctive to express the negative command: Quod tibi non vis, alteri ne feceris.
DR 176. Quod tibi, hoc alteri. ~ Note: This is another succinct statement of the Golden Rule, with the verb implied by not stated: That which you (do) to another, (should be what you would do) for yourself.
DR 176. Accipe quod tuum alterique da suum. ~ Note: Again, there are some words implied but not stated in the compact Latin: Accipe (hoc), quod tuum (est), alterique da suum. Note that suum refers to the person who is "alter" - give to another what is his (or hers).
DR 176. Alterius ne sit, qui suus esse potest. ~ Note: Note the genitive alterius - "ne sit alterius" means something like "do not become another's," i.e. do not become the slave of another, the servant of some other person.
DR 178. Quod scripsi, scripsi. ~ Note: This saying is included by Polydorus in his Adagia, B175.
DR 180. Rem tene; verba sequentur! ~ Note: Note the future tense: sequentur. (It's all a matter of vowels: sequuntur, present indicative; sequantur, present subjunctive; sequenter, future indicative.)