He was checking her out. But Jack is a gentleman and lied about it.

I thought Jack and Kate shared more chemistry. To
each their own.

Kate and Sawyer had not seen each other in three years, they had each moved on. And like you said, Kate made no romantic overture towards Sawyer.

I am not saying Fox is an angel. But I find it interesting that Dom hasn't uttered another word about it, considering all his conviction at the time. Also, Dom has a documented abusive verbal streak when it comes to engaging women. So, the source is highly unreliable.

Absolutely.

You summed it all. Jack was my favorite character for the same reasons that you described him as the best character. I think he is very under appreciated, yet I think he has the most rewatch value in the entire show.

I wholeheartedly disagree. Kate never took any actions to come between them. It was Sawyer looking at Kate, while she is (sadly and wistfully) looking into the distance that triggered the events. If there is anyone to point blame at, it's Sawyer. Kate never made an overture towards Sawyer. And the ending confirms…

Considering that more than three years have passed since the accusations and nothing has surfaced to substantiate them, perhaps it's time to consider that they were a lie. Dom clearly holds some hostility towards Fox and Lilly (he attacked her on twitter too). Maybe it's time to question the accuser.