This one is infamous, a dog dies after his owner Shou Tucker uses alchemy to fuse the dog with his young daughter, Nina, into a chimera which creates an whole new entity arguably unaware of the two beings it used to be. This poor creature is then put out of it's misery by a different character. Also it is established that Tucker had previously attempted the same thing with his wife.
Though not explicitly shown Rose, a 16 yr old girl, is cornered by a group of soldiers, then it fades to black. The next time we see her is almost a year later, and she has a baby. Someone tels Ed that she was taken by a group of soldiers and was never the same.
This one is very missable in that the character's identity is mostly implied and only confirmed (?) through other FMA media. The character is genderless/nonbinary but referred to as he" in the 2003 anime; however "it" is also used (in the manga I believe) and many fans use "they". The character dies near the end. To my knowledge, there are no other confirmed LGBT characters in the 2003 anime.