New yYear's hHoliday

"New Year's Holiday" is a proper noun, and thus should be capitalized.

I spend three days on staying at my girlfriend's house during naround New yYear's holiday.

In the first place, "new year's holiday" is a bit of a weird phrase that isn't really used in English; there's not really a direct translation for 年末年始, but "around New Year's" will cause people to think of the correct time period.

She is seriously addictived to games and was eager to watch RTA in jJapan, which is a streaming event performing time attack race at challengers'where people speedrun their favorite games.

While "time attack" isn't entirely incorrect, "speedrun" is the much more commonly used word for this type of thing.

Challengers make efforts to develop their best record with their consummate skills (incredible feat of skill) and bugs (cheat code), and XXX as welland use of glitches, while simultaneously XXX.

The parentheses come across as a bit unnatural, especially since "cheat codes" aren't really the same as "bugs", and as far as I've seen, typically don't show up in this kind of event. Also, "glitches" is slightly more accurate than "bugs".

Whatever "XXX" is, it's unrelated to the challengers' records, and should be separated. Personally, I dislike the other correction of "wooing the crowed". I'd maybe suggest "drawing in viewers with their banter".　意味的には、「fascinating viewers with their funny talk」の似ているけど、もう少し一般的で、「explaining how fun the game is」も「letting viewers know the exciting points」も「recruiting new challengers」も一応含められています。「banter」というのはそれほど一般的な言葉だからです。

(I can't make up my mind which phrase is suitable in this part. ¶
XXX = recruiting new challengers / explaining how fun the game is / let youting viewers know the exciting points / fascinate bying viewers with their funny talk / demonstrating their skills)

I spendt three days on staying at my girlfriend's house during the new year's holiday.

She is seriously addictived to games and was eager to watch RTA in jJapan, which is a streaming event performingof time attack races of at challengers' favorite game.

Not quite sure what ´time attack race´ is, if this is a known gamer phrase or not. If not, then it´s not very clear so try to reword.

Challengers make efforts to developtry to set their best record withs by displaying consummate skills (incredible feats of skill) and, developing bugs (cheat codes), and XXXwooing the crowd as well.

You can say ´make efforts to´, it´s okay, but its a bit wordy and out of place here.
´Consummate´ is not a word natives would use or see often, but maybe it´s used specficially for these competitions, so that´s fine, just be aware.
In general, you ´set´ a record, you don´t ´develop´ it (though you do develop skills, so perhaps that is moreso what you want to say?)