Ironically, I couldn't go out to see the/my (= both are ok) dentist today and that's due to the intense pain I feel on my gum.

It's very rare that one cancels an appointment at the destist because of a toothache.

Sumo is a (pure) japanese traditional way of wrestling and I'm a fan of it.

Hakuho was about to set a new world record in the long history of Sumo.

My gum still hurts

My gum has still been hurting [ENG/日本語] This syntax is more natural, but I would suggest changing it to "My gum is still hurting," because "has been hurting" almost sounds like it isn't hurting at the moment. Also, when referring to the gum in the mouth, it's usually plural. I guess because you have gums on both the top and the bottom of your mouth. If it's only your top gum hurting, I think it'd be okay to say "gum" in singular form. If it's both top and bottom, I think it'd be better to use gums. I've actually never heard this kind of gum referred to in singular form, so it might be safer just to say gums in plural instead. ^^;

Ironically, I couldn't go out to see [my/the?] dentist today because of intense pain [on/in?] my gum. "My dentist" would suggest that it is the dentist that you see whenever you go to the dentist. "The dentist" might suggest that you don't see this dentist whenever you go. For "intense pain on/in my gum," I guess I would use "in," but either one sounds odd to me. It might be better to just omit "on/in my gum," because the reader can assume from your title that you must be talking about your gums hurting.

It's very strange that one cancels a dentist's appointment because of a toothache. The original sentence is fine, I don't know why the previous corrector changed it. ^^;

Sumo is traditional Japanese wrestling and I'm a big fan. For some reason, "I'm a fan of it" sounds a little odd. "Of it" can easily be omitted without confusing the reader, so you can say "I'm a fan," or to give more reason for your disappointment, you can say, "I'm a big fan."

Hakuho was about to set the best record in the long history of Sumo world in Japan. "To set a record" can probably be considered a set phrase. I would omit the last three words of the sentence, because you already mentioned that Sumo is traditional Japanese wrestling, so mentioning that he would have set a record in the history of Sumo in Japan seems a little redundant. ^^;