2 domains, 1 wp installation

Hi, I read a lot of topics, but now I a bit more confused.
I have a website, not bad indexed, but geotagged in Italy ( mywebsite.it )
I created english translations for my pages (actually accessible from mywebsite.it/english)
I bought a new domain: mywebsite.com Actually it redirects to mywebsite.it
I think it should redirect to mywebsite.it/english for better indexing in English language.
Is it correct? How can I do it ?

Is it possible to show to “English” visitors the new domain in all pages instead of “mywebsite.it/English/pagename” ?
Many thanks