看起来很奇怪 should be perfectly okay as well.
Actually, you need to say the person and the hat.
What is the difference between 看 and 看起来 in this situation?
看 means "look(s)", however, 看起来 means "looks like" or "looks as if".
Ok, ta. But in which case the hover translation has no "right" suggesting both. It's deliberate deception.
看=look as in see, regard. 看寄来=look as in appear.
Is the 的 completely necessary, or can it be dropped in some situations?
There are very rare situations where it can be dropped: for one's own family members when no possibility for confusion exists e.g. "我爸爸说我应该马上回家＂For your own bodyparts. E.g. "我头很疼＂.
Here you cannot omit it.
I wish it were clearer when one uses look, and when look from come. 看 看起来