Adobe Acrobat converts my files from Word to PDF on it's own on my MacBook Air

For starters, Acrobat isn't converting anything. It has simply assigned itself as the default app to open .doc and .docx files, which it then presents like a PDF.

To get the OS to assign them back to Word, right click on a .doc or .docx file-> click on Get Info-> click on the drop down menu where it says "Opens With"-> select Microsoft Word-> click the Change All… button-> click Continue.

You'll have to do the above to .doc and .docx files separately.

As far as Acrobat pestering you about the 7 day trial, each Creative Cloud account allows you to activate the software on two computers, and two only (any combination of Windows or Mac). That it keeps asking sounds like two are already in use, so you can't activate a third. As such, it remains in trial mode.

I have been signing in to Acrobat with my Apple sign in though.

That isn't possible. Apple and Adobe each have their own login servers. Though I'm guessing what you may mean is you're using the same email address as your user account name for both companies.

What you show is a bit confusing for a free Adobe account. It shows items that, according to Adobe's site, are not party of a free account. This is what they show.

Regardless of the mismatch, if this is what you're logged into on your Mac, that's why you can't use Acrobat Pro. It's not part of a free account.

That I know of, you can only be logged into one Adobe account at a time. If you want access to your partner's Creative Cloud apps, you have to login to his or her account.