Learn Hardcore German: Schmerz macht mich stark.

Because Schmerz is an abstract, uncountable noun describing the general concept of pain. German often omits the article with abstract or uncountable nouns when speaking in a general sense, for example Liebe macht glücklich or Hunger ist schlimm.

Could I say Der Schmerz macht mich stark?

Yes. Adding der makes it a specific or personified pain: “The pain (I’m experiencing) makes me strong.” Without the article it’s more generic: “Pain makes me strong.” Both are correct, but the unarticulated form emphasizes the universal idea.

Why is mich in the accusative case?

Because the verb machen (in the sense “to make someone something”) takes its direct object in the accusative. So in jemanden stark machen, the person being made strong is in the accusative: mich.

Why is stark not declined or given an ending?

Here stark is a predicative adjective following the verb and acts as an adjectival complement. Predicative adjectives in German remain in their base form without any case or gender endings.

Could I use the verb stärken instead and say Schmerz stärkt mich?