Galadriel is certainly capable of commanding the Mirror; she says so herself:
'What shall we look for, and what shall we see?' asked Frodo, filled with awe.
'Many things I can command the Mirror to reveal,' she answered, 'and to some I can show what they desire to see. But the Mirror will also show things unbidden, and those are often stranger and more profitable than things which we wish to behold.
Fellowship of the Ring Book II Chapter 7: "The Mirror of Galadriel"
Whether she did or not is unknown, and unclear. If she did force Frodo to see the Eye, her reasons are her own and she never shares them.
If she doesn't, however, there is no in-universe reason for why Frodo sees the Eye; the functioning of the Mirror is mysterious and unpredictable, as Galadriel admits:
What you will see, if you leave the Mirror free to work, I cannot tell. For it shows things that were, and things that are, things that yet may be. But which it is that he sees, even the wisest cannot always tell.
Fellowship of the Ring Book II Chapter 7: "The Mirror of Galadriel"
However, I question why Frodo shouldn't have seen a metaphorical depiction. What Frodo sees isn't Sauron himself, but Sauron's search for him; consider what he actually sees:
Then the Eye began to rove, searching this way and that; and Frodo knew with certainty and horror that among the many things that it sought he himself was one.
Fellowship of the Ring Book II Chapter 7: "The Mirror of Galadriel"
That's really the most important thing Frodo could have seen from the mirror: not Sauron himself, but Sauron's desire for the Ring; it's hard to show something like that non-metaphorically.
There's no evidence that Frodo seeing the Eye has anything to do with being able to perceive the other Ringbearers. In fact, Frodo can't see the thoughts and minds of the other Ringbearers (not yet, anyway. If he put some work in, perhaps he could); unless put in a situation where he can actually see their Rings, he doesn't even know who they are. What he can see is the Rings themselves. Galadriel answers this also, in more context given around one of the quotes in the question:
'I would ask one thing before we go,' said Frodo, 'a thing which I often meant to ask Gandalf in Rivendell. I am permitted to wear the One Ring: why cannot I see all the others and know the thoughts of those that wear them?'
'You have not tried,' she said. 'Only thrice have you set the Ring upon your finger since you knew what you possessed. Do not try! It would destroy you. Did not Gandalf tell you that the rings give power according to the measure of each possessor? Before you could use that power you would need to become far stronger, and to train your will to the domination of others. Yet even so, as Ring-bearer and as one that has borne it on finger and seen that which is hidden, your sight is grown keener. You have perceived my thought more clearly than many that are accounted wise. You saw the Eye of him that holds the Seven and the Nine. And did you not see and recognize the ring upon my finger?
Fellowship of the Ring Book II Chapter 7: "The Mirror of Galadriel"