Both North and South Vietnam claimed they were the legitimate government of Vietnam. Both saw the other as an "illegitimate regime" occupying their half of the country. The South wanted reunification but under a capitalist and anti-communist government. Meanwhile, the North wanted reunification under a communist government. The situation is similar to how both North and South Korea view each other.

As for the second answer, the situation is a lot more complex. Anti-communist support was quite strong in the cities and urban areas of South Vietnam. In the rural areas, it was more mixed with some supporting communism and others neutral.

Since both North and South Vietnam were quite authoritarian, so it is difficult to get any accurate info on this matter. If an unrigged referendum was held in South Vietnam on this issue, it would have been extremely divisive. Ho Chi Minh and the communists had earned a lot of support in their fight against Japan and France. On the other hand, there also many Vietnamese nationalists that did not support communism at all.

Although we won't know the results of this referendum, we can look at how the South Vietnamese people reacted to the Tet Offensive in 1968. In 1968 on the Vietnamese New Year of Tet, the Viet Cong (supported by North Vietnam) launched a massive offensive in the South. The goal was to trigger a national uprising in the South. It was hoped that South Vietnamese soldiers and civilians would rally in support of the Viet Cong against the South's government and US military. The offensive resulted in the complete opposite effect. Civilians ran towards the South Vietnamese and US forces for protection. Despite being made up of mostly conscripts, the South Vietnamese army fought back against the Viet Cong rather than defect.

Here’s a breakdown and context:

South Vietnam was still basically a French colony and controlled by French people, too. They followed Catholicism and had a large army, of course being a military junta, and somewhat having propaganda. North Vietnam was supported by the Chinese and was an atheist state, practically a group of rebels led by a certain Ho Chi Minh. South Vietnam disliked north Vietnam for obvious reasons mainly due to U.S. control which supported its army. Opinions before the war were practically the same except they more thought of themselves as a race. A bit more context is that most Vietnamese people had family members who’s lives were lost in the war due to the USA prompting the Viet Cong.

As for the second question, it’s mixed. The southerners would definitely wanted a better unified Vietnam, however opinions would be mixed on who would control it. Probably not. But as considering the long run they would have to be forced to join otherwise and most southern Vietnamese would go against it thinking their country would become the next South Korea. Hope this helps