Opened .docx file with Apple pages for an apparent job screening. Danger?

If you double-clicked on that Word document and got a System warning of any kind or an installer wanting your admin password, then Cancel that dialog and delete the Word document. It might have been an installer masquerading as a Word document. If you canceled and deleted the document, you have nothing to worry about.

Opening a Word .docx file from an unknown site in Pages is no guarantee that a) the original formatting of the Word document in Word, will be preserved in Pages; or b) Pages does not support Word form fields; and c) any vermin in that Word document, even if you double-clicked it to open in Pages cannot get installed into the operating system as it is on a read-only, code-signed System volume.

Something you should know about consulting sites like this is that they are résumé collectors for potential future bids that they may or may not win, and your PDF will likely collect dust on their site. The phone number should have been a red flag not to send them anything.

Your best bet is to provide an electronic and printed version of your résumé to someone you know well in your industry who can hand-carry it to a hiring decision-maker. Harder than it sounds as most people don't won't to risk their careers on this activity. It also avoids the omnipresent application tracking software on recruiting/corporate websites that reject 94% of all submissions and the human resource staff that usually have no clue what they are reading in the remaining 6%. They are looking for keywords that match the job offer and not even the best match for the job. This frustrates the living daylights out of the managers who want more from an applicant than what HR provides them. But… corporate hiring policies have shifted away from hiring managers to HR protocols that control this process.