In American, "I see" is often used to mean "I understand." Do Hungarians do that?
Not really. There are many expressions that are something similar but most of them belong to the slang and colloquial language. "Vettem" (lit: "I got it / bought it / received it") is something like "Roger" or "ten-four" in aviation or CB-slang. "Vágom" is a youth slang. There are many more but I don't remember "látom" in this meaning and in daily use. There are poetic and literature occurences, though.
In Hungarian they tend to use "Értem" (I understand it) where we'd say "I see".
He/She is seeing is technically correct. If you were rejected, you may want to report it.
She can see is frequent in spoken language, but it literally means "Ő képes látni" and in this case "ő lát" is an abbreviated form. Without the context it is right, too.