Learn Hardcore Russian: Новые методы исследования обсуждаются на лекции по физике. - New research methods are being discussed in the physics lecture.

Because методы (“methods”) is the subject of the sentence and is in the nominative plural. Adjectives agree with their nouns in gender, number, and case. Here новые is masculine-animate (since “метод” is a masculine noun) plural, nominative case, matching методы.

What case and role does исследования have in this sentence?

исследования is the genitive singular of исследование (“research”). In Russian, to say “methods of research,” you put the second noun in the genitive. So методы исследования literally means “methods of research.”

What does обсуждаются mean, and why does it have the suffix -ся?

обсуждаются is the third person plural, present tense, imperfective passive form of обсуждать (“to discuss”). The -ся suffix turns the active verb into a reflexive/passive construction. Here it means “are being discussed” or “are discussed” (habitually or currently) rather than “discuss” (active voice).

Why is на лекции in the prepositional case?

The preposition на when it indicates a location or event (“at,” “on”) generally takes the prepositional case. лекция in the prepositional singular becomes лекции, so на лекции = “at the lecture.”

What is the function and case of по физике?

по here means “on” or “in the subject of,” and it governs the dative case. физика (nominative) becomes физике in the dative. Together по физике means “in physics” or “in the physics course/lecture.”

How can I express the idea that the discussion has already been completed (i.e. use the perfective/passive past)?

You can use the past passive participle with an auxiliary verb:

Новые методы исследования на лекции по физике были обсуждены.
Here были обсуждены is the past passive perfective, meaning “were discussed” (completed action).

I want to generalize to all physics lectures, not just one. How do I change the sentence?

Make лекции plural in both the prepositional and the preposition на:

На лекциях по физике обсуждаются новые методы исследования.
Now it reads “In (all) physics lectures, new research methods are discussed.”

How do I specify who is doing the discussing, or rephrase it in the active voice?

1) To add an agent in the passive, use the instrumental case with кем (“by whom”) or just add the doer in the instrumental:

Новые методы исследования обсуждаются на лекции по физике профессором Ивановым.
(― “...are being discussed by Professor Ivanov.”)
2) In the active voice, simply name the subject and drop -ся:
Преподаватель (Иванов) на лекции по физике обсуждает новые методы исследования.
(― “The lecturer (Ivanov) discusses new research methods at the physics lecture.”)