So, any opinions on how should I fix my UI?, and any other things to fix if there is?
You're allowed to use their logos, so that's not a problem:
But, then you use a very similar display mechanism for the questions:
There's no reason why your UI should look so similar to their UI. It may cause a little confusion; but only to the newly initiated user. For the information you're displaying, you could keep it clean and still include everything they do.
As far as the timestamps used on the posts, Stack Overflow doesn't have a monopoly on how to display time, so I wouldn't worry about that, but I would change the question display interface itself.
To fix your UI, you could:
This is exactly the type of issue I was afraid would happen:
Do you intend to pretend to be a SOInc product? Do you think a reasonable person would believe your site is a SOInc product?
I can't give you a concrete answer, but I do wish the guidelines didn't make you overly concerned.
However, I can suggest trying to apply your own branding. A masthead across the top of the page is relatively easy, for example. Clearly set yourself apart so it's obvious you're not a SOInc product.