According to the Lost Encyclopedia, the surname Rom was given by the Others after Ethan defected from the DHARMA, so Amy's surname wasn't Rom.
Is an Other
Amy was originally an Other, and it was her Ankh necklace.
Amy is a spy for the Hostiles. That is why she was so quick to support the decision to kill Sayid (she knew he wasn't a hostile). She runs away with Ethan before the Purge.
The theory that Amy is a spy or mole is also supported by her and Paul's possession of the Ankh, which has been linked to the Others/Hostiles. Also, the tunnel that goes from the Temple to the DHARMA camp has its entrance in the basement of Horace and Amy's cabin. This was likely how Amy and Paul ended up outside the perimiter of the DHARMA camp when they were found by Sawyer and Juliet in 1974. The Hostiles who were killed by Sawyer were not going to kill Amy and Paul, but had placed the bags over their heads to take them to a secret location to meet with the Others, possibly Richard.
Amy is the spy, but these two Others in 5x08 didn't know about it. That's why they killed Paul.
Amy is Amelia. She is the right age and is Ethan's (who defected with Ben to the Others) mother.
Amy died in the purge according to the Lost Encyclopedia so can't be Amelia or any other modern day other.
Why in Other's territory
May have been doing something they shouldn't have been doing (espionage, sabotage?) in the Others' territory. If they didn't have some kind of plan in mind, they were simply wandering around in the middle of hostile territory for no good reason. This would explain why the Others kill Paul and attempt to kidnap Amy despite the Truce. Amy also cries out that they were "just talking" as the Others kidnap her; perhaps she was trying to cover something up.
The picnic was within a ring of flowers, potentially an especially fertile spot, sacred to the Others.
This was why Paul was wearing the Ankh symbol - an Egyptian symbol of life, specifically the giving of life, thought to include the act of conception. Paul was no hippie flower child - he was trying his best to be fertile for Amy.