Learn Arabic (Modern Standard): كانت نسبة المشاركة عالية.

The verb must agree with its subject in gender. The subject here is the first part of the Idafa phrase نسبة المشاركة (the participation rate). Since نسبة ends in a taa marbuta (ة), it is feminine, so we need the feminine verb كانت.

In English it says 'The participation rate', so why doesn't the Arabic word نسبة have the definite article الـ attached to it?

This phrase is an Idafa (a genitive construct), literally translating to 'rate of the participation'. In Arabic, the first word of an Idafa never takes the prefix الـ. It automatically becomes definite because the second word, المشاركة, has الـ.

If I were to read this sentence with full formal vowel endings (tashkeel), what case would the word عالية take?

It would take the accusative case (mansub) and be pronounced عاليةً (aliyatan). This is because it is the predicate (the descriptive part) of the verb كان (was). The verb كان leaves its subject in the nominative case (marfu') but forces its predicate into the accusative case.

I recognize the root in المشاركة. What verb form does this noun come from?

It is the verbal noun (masdar) of the Form III verb شارك (to participate or to share). Verbal nouns for Form III often follow the pattern 'mufa'alah', which gives us مشاركة.