Sharingan genjutsu, yes. Normal genjutsu, prolly not
Sharingan genjutsu is better than normal genjutsu, both of em yes, tsukyomi might resist
yes, it is disturbance in chakra, but just when the user of sage mode is not moving, becose natural chakra can be absorbed only when you are staying. if the user of sage mode is moving he can't absorb chakra and he can be traped in genjutsu .