Psychic Damage hasn't visible effect. They don't know about damaging.No, the target knows they took damage. They're not going to just stand there and keep taking it. Just running away thwarts your dastardly plan, and, in social engagements, knowing there's a hidden assassin using mind magic is going to get bad quickly for anyone that could be considered suspicious -- like a sorcerer that's just come visiting. Your awareness of any broader context is lacking.
The Mind Spike knows the target location for 1 hour.
On a failed save, you also always know the target's location until the spell ends, but only while the two of you are on the same plane of existence. While you have this knowledge, the target can’t become hidden from you, and if it’s invisible, it gains no benefit from that condition against you.