Whoever objects to me going to the convention ought to state

To express a feeling or desire, we should use possessive pronouns, so option D is correct.

Please correct me if i am wrong.

What will you say about- "He loves me." or "He loves my."

Here he is expressing his feeling, still object pronoun is used.

It also depends on the tone of the sentence. The above sentence uses a strong tone to express the feelings of the person, so the use of possessive pronoun is more appropriate than the object pronoun.

For sentences of this nature (read: formal/uncommon), it's best to rephrase it into a "normal" structure to identify the necessary pronouns and verbs.
Whoever <object/objects> to <me/my> going to the convention ought to state his position promptly --> If he objects to me going (to the convention), he ought to state his position promptly.

Because it's HE, you should always stick with WHO (only use WHOM when the revised sentence pronoun is HIM/HER)... and because I am the one going to the convention, the correct pronoun in the original sentence should be MY.

He/She --> Who
Him/Her --> Whom

He/She/ I --> His/Her/My

Ans: D

D33T
We can't use the presence of "I" to justify a possessive ("my"). The two really have nothing to do with each other. I could say "I want you to invite me" or "I want you to invite my cousin." The issue is whether the pronoun is serving as the object of a verb (me) or as a modifier (my). In this sentence, no one is objecting to ME. They're objecting to the idea that I will go to the convention. So they object to my attendance, or in this case, "my going."

This is one of those usages that is formally correct but does not reflect how most of us speak in daily life. In real life, most people would probably say "Do you mind me tagging along tonight?" Or "I'm bothered by Darren taking charge," but technically, we should say "MY TAGGING" and "DARREN'S TAKING." Certainly, some of us speak this way, especially if we are trained in SC, but this may sound weird even to many fluent and/or native speakers of English.