While the succubus is in the ethereal plane, can you see her?
Yes, if you are on the ethereal plane and near enough to see her through all of the mists and such. If you are not on the ethereal plane -- no. The exception to that case is use of a device or spell that allows you to see into the ethereal plane. (The two spells true seeing & see invisible would apply, per SRD p. 198, and magical items like Robe of Eyes or Wand of Enemy Detection).
Does her charm ability works in the material plane while she is in the ethereal plane?
No. The ability to apply that effect does not specifically call out a cross planar capability. You could make the argument that when using an item to see into the ethereal plane, and be in telepathic contact with the succubus, she could charm you. That's worth discussing with your DM.
Note this from Beautiful Corrupters (MM p. 284):
Inevitably, the fiend enters the mortal realm in tempting form to directly influence a creature's actions.
This implies that for all of the RP potential in the previous passage about influencing a target to do things via dream suggestion, the succubus must cross the planar boundary in order to have direct impact on the character/creature. You might call the dream influence a preparation step.
Can she travel to the deep ethereal?
There is no reason why she can't, though whether or not she'd want to is a matter for the DM to derive.