Is there a name for the experience/ phenomenon of dreaming while being partially awake, being able to decide to wake up when dreaming, having control of the self character in a dream, etc?
A dream where you are aware you are dreaming is called a lucid dream (see also La Berge et al 1981). Lucid dreams can have varying level of control; in La Berge et al 1981 subjects were able to report by eye movements and fist clenches when they were lucid dreaming, and were verified to be in REM sleep when this occurred.
Being aware of your outside surroundings, on the other hand, is called being awake and is not the same as dreaming but could be considered daydreaming.
Some people report that they often have lucid dreams, and lucid dreaming is more common in some patient populations like narcoleptics (Dodet et al 2015).
There are methods to encourage lucid dreaming if one wants to experience them (La Berge 1980, Zadra et al 1992).
Dodet, P., Chavez, M., Leu-Semenescu, S., Golmard, J. L., & Arnulf, I. (2015). Lucid dreaming in narcolepsy. Sleep, 38(3), 487-497.
La Berge, S. P. (1980). Lucid dreaming as a learnable skill: A case study. Perceptual and Motor Skills, 51(3_suppl2), 1039-1042.
La Berge, S. P., Nagel, L. E., Dement, W. C., & Zarcone Jr, V. P. (1981). Lucid dreaming verified by volitional communication during REM sleep. Perceptual and motor skills, 52(3), 727-732.
Zadra, A. L., Donderi, D. C., & Pihl, R. O. (1992). Efficacy of lucid dream induction for lucid and non-lucid dreamers. Dreaming, 2(2), 85.