Short answers, pulled from the story.
Arhat comes from the Sanskrit root meaning "to deserve" or "to merit." Related forms include arha (deserving) and arhita (honored). Tibetan translators rendered it by meaning as dgra bcom pa, or "one who has destroyed the foes of afflictions," understanding arhat as ari-hanta.
It depends on the school. Theravada Buddhism considers the Buddha himself to be an arhat, and treats the two as closely similar. Mahayana Buddhism places arhats below buddhas in a clear hierarchy, arguing that the arhat's path is motivated by personal liberation rather than the altruistic orientation of a bodhisattva.
An arhat who is still alive has attained nirvana with residue: the five aggregates (form, sensation, perception, mental formations, consciousness) continue to function, sustained by physical vitality. At death, when the body disintegrates, those aggregates cease entirely, producing nirvana without residue, also called parinirvana.
The Eighteen Arhats are a specific group in Mahayana tradition, each with a distinct name and personality, described as remaining in the world to await the future Buddha Maitreya. The Chinese monk Guanxiu painted the first famous portraits of this group in 891 CE and donated them to Shengyin Temple in Qiantang (modern Hangzhou).
Yes. The Kāśyapīya school held that arhats had not fully eliminated desire, that their perfection was incomplete, and that relapse was possible. The Sarvāstivādins took a similar position, and both the Mahāsāṃghika branch and the Caitika schools also viewed arhats as fallible.
Mahayana texts describe sixty novice bodhisattvas who accidentally attained arhatship because they lacked the abilities of prajnaparamita and skillful means. One text compares them to a giant bird without wings falling from Mount Sumeru. Those who reach arhatship and fail to take up the bodhisattva path are said to enter a deep samadhi of emptiness, from which they will be roused and taught when ready.