Meghla performed

  1. Meghla performed good/well on the exam.
    Good and well both can be use as a adjective. Then why is only “well” correct here?
  2. After going to the dance, Mary was really tired.
    What is the name of the bold part? Is it modifier or what?
    In addition, in what part of speech we can classify it?

Yes, “good” and “well” can be adjectives. “I feel good” = I am happy; “I feel well” = I am in good health.

But, I believe, only “well” can be an adverb in standard English. How did Meghla perform? She performed well. (An adverb that tells you how she performed.)

In popular speech, we often hear, “The baseball team played real [“really”] good [“well”] last night.” That sentence would upset your teachers if you wrote that in a formal report.

I think that most books for high school (secondary) students would simply say that “after going to the dance” is a prepositional phrase that modifies the verb “was.” It answers the question: When WAS Mary really tired?

The prepositional phrase consists of:

  1. after = the preposition
  2. going = a gerund, which --as you know – is a verb form being used as a noun.
  3. to the dance = a prepositional phrase that modifies “going.” It answers the question: Going where?

Good and well both can be use as a adjective.
Is this sentence correct?

Not quite:
(The words) ‘good’ and ‘both’ can be used as adjectives.