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.
The Ethereal Plane is another dimension, described in more detail on Pages 48 and 49 of the DMG. It may help you to replace the term "plane" with "dimension." The ethereal plane borders several other planes, and is often described as a muted, foggy outline of whatever exists on the plane that it borders.
So, take for example that you are in a corridor inside of Castle Ravenloft. There is a spot in the ethereal plane which looks exactly the same as the castle corridor, except it's all fuzzy looking and grey and foggy. Any creature on the ethereal plane is invisible to the stuff in the castle corridor, but the creatures in the corridor are visible to the ethereal denizens. Creatures on the ethereal plane also feel no gravity, and can move through walls easily.
The ethereal plane is used very often then for travel, as well as espionage, since travelers can move through walls, as well as spy on people on the material plane without worry of being discovered. Ghosts and other such incorporial undead can move into this plane at will (which is what allows them to turn invisible).
If a character can use true sight, they can see into this plane. So if anything is spying on them using this plane, they will see it. They can also see ghosts and other such spirits that try to hide from them by moving into this plane. They will also see things that were hidden in the ethereal plane for safe keeping.
As for the effect of this on your campign, that really depends on how much you use alternate dimensions as a DM. If you don't use the ethereal plane for very much, the spell just means that the PC can follow ghosts and other such things, as you have noted. If you like to put spies in there, or other such hidden things "stashed away in the fourth dimension", then the PCs will notice those things. The effect is pretty minimal, since your players would acctually have to shift themselves into the ethereal plane in order to affect anything you put on that plane.
As noted by Andrewk, Ravenloft is filled to the brim with trapped ghosts. Even if they don't directly interact with the players, it can be a powerful narrative/aesthetic device for the PCs to see so many souls in the ethereal plane as they move through the castle.
Best Answer
No, not unless a spell specifically allow it to cross planar boundaries
Etherealness says:
Unless the caster has a means to transport their effect to the Ethereal Plane, all spell effects that they create will be in the Material Plane. And since a creature under the effects of etherealness is immune to effects not in the Ethereal Plane they would not be affected by them.
Additional support comes from the part you already quoted:
True seeing might give a creature the ability to see you, but it does not magically give them the ability to interact with you. You are still on a separate plane. Thus, they are not able to target you with effects for this reason as well as it is a type of interaction.
Spell which target by sight still have to meet other requirements
As a related aside: when a spell says that you must be able to see the target, it does not mean that this is the only requirement that the spell has for being able to target something. All spells, for example, must have a clear path to the target (unless something in the spell effect says otherwise) by default.
Though nothing says it explicitly in the rules, it seems the most logical ruling would be that you do not have a clear path to things on a separate plane from yourself. Spell effects cannot cross planar boundaries unless they explicitly say so. Adding a requirement that you must be able to see the target does not bypass these other considerations.
Jeremy Crawford also unofficially supports this idea in a Tweet by saying that another plane is an infinite distance away.
Being an infinite distance away means that any spell with a limited range will fail.