Can certain moves damage other simultaneous attackers in Pokémon Go?

So far there is no evidence that trainers attacking a gym interact directly at all. They can all do damage to the defending pokemon (allowing it to be defeated much faster), and the defending pokemon will do damage to all attackers at a constant rate independent of how many there are, but there is no indication that attackers can hurt each other.

Moves that would normally have special effects, such as status effects, in the regular series do only damage in Pokemon Go. This would imply that many other "side effects" of attacks have also been removed.

Pokemon Go has removed a lot of features that were in other Pokemon games, and I believe all the mechanics of double-battles are included in that set of removed features.

No they don't effect other pokemon. Other than elemental varients no moves have any special effects whatsoever.