It's generally accepted as canonical that Roy Mustang refuses to make any moves while playing chess which would require him to sacrifice his queen (which is the piece to which he compares his devotedadjutant, Lieutenant Hawkeye). This stems from how when comparing his moves to chess, he makes a point about the importance of Hawkeye by comparing her to the Queen piece.
2003 Anime
Two of the most common names for our-world Winry and Riza are "Wendy" and "Elizabeth".
Envy's original name is always "William Elric" in fanfics. A lot of fans think it's official but it isn't.
The alter counterparts to Sheska, and to a lesser extent Winry, are often depicted as Jewish in post-COS fanfics.
It's unlikely in canon due to one being German and the other British, but several fans depict Alfons and alter Ed as siblings. It's also common to depict alter Ed and alter Envy (using his original form's looks) as half-brothers.
It's never stated how many soldiers raped Rose but fans insist it was more than one, often a gang-rape.
Riza keeps her back tattoo from the manga in many fanworks. It's never shown, but it's never debunked either.
Alters are often depicted as the opposite of their Amestris versions. For example, alter Winry is usually depicted as a softspoken Girly Girl who Ed sees as a Replacement Goldfish.
Alter Roy and alter Riza are close friends (or more) because fans can't imagine one without the other. This is supported by a piece of artwork depicting them together despite the fact there's no way Riza was in the military.
Edward and Winry have mutual crushes on one another. The 2003 anime is intentionally vague on whether they're Like Brother and Sister or whether they're Implied Love Interests, with signs on both ends.
Rose's child is a boy. It's heavily implied but never stated.
2003-universe fanfiction often assumes the existence of Xing, the Northern Wall of Briggs, and other locations from the manga and Brotherhood that don't appear in the 2003 series' tighter story. It's not uncommon to see characters like Ling or Olivier in post-series fanfic (especially Fix Fic that put Ed and Al back in Amestris). There have even been a few post-Shamballa fics that have Al fall for our world's version of May Chang. In the anime, a far-east country is mentioned, but it's never described as being Xing and is unnamed since by the time it was introduced, the anime has gone into its own continuity.
Both
The idea that Ishval was an independent country Amestris tricked into attacking, so that Amestris could invade and conquer them. Actually Ishval was just a region of Amestris that the government tricked into rebelling so that they could crush and massacre them. This is made clear in the manga when several scenes from the Ishval Massacre reveal that the Amestrian army still had Ishvalan soldiers up until the 7th year of the rebellion, and that they only killed the pure blood Ishvalans (and possibly the half-bloods) when they needed souls for a Philosopher's stone. Scar's brother also tells Scar that Amestris allows the Ishvalans to worship Ishvala. In addition, the Amestrian government is currently at war with 4 other nations, and is a military dictatorship, like they would need a Batman Gambit just to invade a 5th.
It is generally accepted that Edward's pants are leather, despite the fact that they would be, as Vic Mignogna points out, "hard to fight in. Ed's gotta be smarter than that." Conversely, as Vic does not write or draw for the series, it is hard to determine the exact material of Ed's pants unless Arakawa-sensei decides to enlighten us for herself.
There was a generally accepted theory that Roy Mustang, because of his more East Asian looks and the fact that he was adopted, was an orphan from Xing (or the eastern nations mentioned in the 2003 anime). However, this theory has been altered slightly with the reveal that Madame Christmas is not only Roy's adoptive mother but also his paternal aunt. More fans now lean toward the idea that he is of partial Xingese ancestry.
In fanfics depicting an adult Elicia, she will inevitably follow Hughes' footsteps and join the military.
Dead Guy Junior is very common in fan-works. Rose's son is usually named after Edward. If Edward stays on our side of the Gate, his son will often be named "Roy" and his daughter either "Trisha" or "Winry" (the name "Trisha" is also often used for the manga with his and Winry's daughter). “Nina” for Ed’s daughter and “Maes” for Roy and Riza’s hypothetical son are also common.
Many fan-works make Ishvalans into a full-blown Fantasy Counterpart Religion version of Islam. This includes Arabic names and Muslim traditions.
Conversely, as Vic does not write or draw for the series, it is hard to determine the exact material of Ed's pants unless Arakawa-sensei decides to enlighten us for herself.