They played a tremendous role, not just in the war(ARVN suffered 250,000+ casualties), but more importantly, in the politics of South Vietnam. It was ARVN mechinations in 1963 and 1964 that led to the US ground presence in South Vietnam, as top South Vietnamese generals increasingly grew more focused on fighting each other than the enemy. As time went on during the Johnson years, the Americans insisted on doing more and more of the fighting anyway.
Depends. They were largely ineffectual in their ability to suppress the VC in a counterinsurgency role prior to 1968. For all intents and purposes by that time, the whole rural countryside was under Viet Cong administration, in spite of increasing amounts of American men and material being deployed to the country. To be fair, a lot of this had to do with the cycle of constant coups that marked the post-Diem era, but the problems extended before then, as seen in the Battle of Ap Bac. This was exacerbated by the fact that American trainers of the time had little to no practical experience in counterinsurgency efforts, particularly compared to colonial powers such as the British in Malaya and Kenya. Korea was and is often cited as a counterpoint to Vietnam, but missed is the fact that the ROKA in South Korea had only one border to defend, snowy and hilly, against a conventional Soviet-esque North Korean Army. The experience was of little use in Vietnam, which had long, winding jungle borders with Cambodia and Laos that were impossible to defend.
After the Tet Offensive, the nature of the war changed to a more conventional one with the PAVN becoming the primary enemy rather than VC. As such, while bearing a superficial similarity to Kennedy era policies stressing a primary South Vietnamese role in their own, Nixon's Vietnamization policy emphasized training ARVN for conventional rather than counterinsurgency warfare. During this time period, ARVN didn't do too badly when they had American air cover and backup: the 1970 Cambodian Campaign and the 1972 Easter Offensive being examples. But when they didn't, such as during Lam Son 719 or the final 1975 Northern Offensive, they fell apart like a rotten fruit. The ARVN gave little confidence to the South Vietnamese people in 1973 and 1974 of being able to defend their country, exacerbated by the cuts in the aid from the US and the rampant corruption of top levels of the ARVN. No matter how bravely the ARVN soldier fought, from beginning to end, they consistently were let down by their officers. Thus, they could do little to rally the people to fight in 1975.
Neil Davis is an Australian photo journalist who covered the Vietnam war. He went on patrols with ARVN units and thought they were quite good. His take on it was that the American units were more concerned with getting back to base and eating pizza. Read his biography more than ten years ago but remember it was great. The part about the Cambodian cannibals who scared the shit out of the Khmer Rouge was something else. Read that while I was in Phnom Phenh.