Many different Buddhist words are translated as "insight" or "enlightenment" (the latter being a terrible Victorian choice of word). The following list is by no means exhaustive:
Vipassanā is a Pali word derived from the older prefix "vi-" meaning "special", and the verbal root "-passanā" meaning "seeing". It is often translated as "insight" or "clear-seeing". But the "vi" in vipassanā has many possible meanings, it could mean to '[see] into', '[see] through' or to '[see] in a special way.'
A synonym for vipassanā is paccakkha "perceptible to the senses" (Pāli; Sanskrit: pratyakṣa), literally "before the eyes," which refers to direct experiential perception. Thus, the type of seeing denoted by vipassanā is that of direct perception, as opposed to knowledge derived from reasoning or argumentation.
Kenshō (見性) is a Japanese term from the Zen tradition. Ken means "seeing", shō means "nature, essence". It is usually translated as "seeing one's (true) nature", that is, the Buddha-nature or nature of mind. Kenshō is an initial insight or awakening, not full Buddhahood. It is to be followed by further training to deepen this insight, and learn to express it in daily life.
Satori (悟り) is a Japanese Buddhist term for awakening, "comprehension; understanding" (thus: insight)
Satori and kenshō are commonly translated as enlightenment, a word that is also used to translate bodhi, prajñā and Buddhahood.
Bodhi (Sanskrit, Pāli), from the verbal root budd, "to awaken", "to understand", means literally "to have woken up and understood".
Prajñā : In Theravada Buddhism pannā (Pali) means "understanding", "wisdom", "insight". In Mahayana Buddhism Prajna (Sanskrit) means "insight" or "wisdom", and entails insight into sunyata. The attainment of this insight is often seen as the attainment of "enlightenment".
Wu (Chinese: 悟) is a concept of awareness, consciousness, or spiritual enlightenment borrowed from the Chinese folk religion.
Many of these terms have a gradual quality to them, implying that insight in most personal cases is a journey (like the Buddha's own journey). Even in Chan/Zen (often called the Sudden Teaching) the "sudden" refers to the insight "striking like lightning" after considerable time walking the path. This is perhaps best illustrated by the "Oxherding Pictures" detailing the stages of insight.
Question 2: yes, but this is generally not regarded as the end of practice. This is generally associated with the 9th step in the "Oxherding Pictures":
Reaching the Source
Too many steps have been taken
returning to the root and the source.
Better to have been blind and deaf
from the beginning!
Dwelling in one's true abode,
unconcerned with and without -
The river flows tranquilly on
and the flowers are red
The final step is to learn to apply this insight in daily life (i.e. to conceptualize it) to help others. You can interpret this in two ways:
a) If this insight was purely non-conceptual, no sutras or books from Masters would exist to "point at the moon".
b) All writing and conceptualization is allegorical, i.e. "pointing to a reflection of the moon in a pond"