Learn English – the difference between ‘despite that’ and ‘despite the fact that’

There's no difference in meaning.

Notwithstanding and despite imply "in spite of" and have been used in the sentences as prepositions. The only difference is that the use of notwithstanding is formal. Moreover, notwithstanding can also be used as an adverb such as He doesn't want me there, but I am going, notwithstanding.

As per @John's answer, #1 is a fairly unlikely statement. But it would always be understood to mean there was something unusual about his eye, and she was (probably, rudely) staring/gawping at it.

But #2 simply isn't idiomatically credible with at - and although you might occasionally come across She stared him in the eye, the idiomatic standard is She looked him in the eye, as you can see from that NGram. For all practical purposes they both mean the same (she looked directly and intently into his eyes, not just at his face).

So even if you might think you want the "looking intently" connotations of stare rather than the more general-purpose look, it's not really a good idea to couple this with in the eye. That's because failure to use standard phrasing will probably simply distract readers, rather than make them pay attention to your attempt to be more "precise".

Note that to look someone in the eye nearly always implies being honest and/or without fear or shame.