Monsters only
You are quoting a passage from the Monster Manual that specifically talks about Monsters within it. There is no reason to apply those telepathic abilities to a PC without an equivalent from the DMG or PHB or another character based reference.
In addition, the Sage Advice clarification on Awakened Mind states:
In contrast, the telepathy ability that some monsters have (MM, 9) does make two-way communication possible.
This clearly differentiates the Monster telepathic ability from others.
Responding to Monster telepathy
Yes, a PC can respond to a Monster using telepathy as stated in the quote provided in the question (MM, 9)
A creature without telepathy can receive and respond to telepathic messages but can't initiate or terminate a telepathic conversation.
Yes, you can use Blink to reappear on the other side of a wall.
At the start of you next turn, and when the spell ends if you are on the Ethereal Plane, you return to an unoccupied space of your choice that you can see within 10 feet of the space you vanished from. (PHB pg. 219)
Notice how this does not say "...from within 10 feet of the space you vanished from." or "...that you can see from the space you vanished from."
Therefore I interpret the wording as "any space within 10 feet of the casting origin you can currently see."
A traveler on the Ethereal Plane is invisible and utterly silent to someone on the overlapped plane, and solid objects on the overlapped plane don't hamper the movement of a creature in the Border Ethereal. The exceptions are certain magical effects (including anything made of magical force) and living beings. This makes the Ethereal Plane ideal for reconnaissance, spying on opponents, and moving around without being detected. The Ethereal Plane also disobeys the laws of gravity; a creature there can move up and down as easily as walking. (DMG pg. 48, emphasis mine)
So you cast Blink close to a wall, roll over 11, move to other side of the wall and, since you are still within 10 feet of your starting location, you return to the material at the start of your next turn where you are currently standing (Or some other square within 10 feet of the origin).
Be sure you already have some idea as to what's on the other side of the wall; You do not want to travel into the wall just to find out it is too wide and pop back into the material while still inside!
Note: 10 feet in DnD is only 2 squares, so you would have to cast Blink right next to the wall.
Best Answer
No, it can't.
Though the Aboleth can use telepathy from the projected image, that's referring to the "Telepathy 120ft." communication ability as listed in the languages section of its stat block - so, the Aboleth can telepathically converse at this increased range, and its Probing Telepathy special ability works as normal, since that doesn't specify a range. However, in order to use its magical enslaving ability, the target must be within 30ft of the real Aboleth.
You may want to note that nothing in the description of the Aboleth's Enslave action suggests that telepathy is required to produce the effect - though that would certainly be in keeping with the Aboleth's theme. As written, even a creature which is somehow prevented from sending or receiving telepathic communication is still vulnerable to the Aboleth's enslavement (presuming that whatever effect they are under doesn't also preclude being charmed).