If I use my Apple ID on my new Mac, does it automatically transfer photos?

You can decide what things to sync with iCloud.

Set up iCloud on your Mac - Apple Support

If you are using iCloud Photos on your phone, and you turn on iCloud Photos on your Mac, then all of the photos and videos that are taking up space on your phone, and in iCloud, will also take up space on your Mac. You can turn the Mac's switch for iCloud Photos off, while leaving the ones for Contacts, Calendars, Notes, and Reminders turned on. Then the Mac's main Photos Library will operate independently from your iPhone's Photos Library (letting you keep it empty, or near-empty).

Since your Mac is not a cell phone, it cannot send or receive SMS/MMS text messages, or send or take phone calls, without the aid of your iPhone. You may find the following Support documents useful for setting that up:

Continuity features and requirements on Apple devices - Apple Support

Forward SMS/MMS text messages from your iPhone to other devices - Apple Support

Make and receive phone calls on Mac or iPad - Apple Support

Yes, there are 2 ways to do that depending on whether or not you are using iCloud for photos on your phone.

If you are using iCloud with Photos on your phone -

On your new Mac, in System Settings, click on the large icon that has your name with the words "Apple ID" below your name. In the resulting window, click iCloud, then under "Apps using iCloud" turn off Photos. That way your phone's photos, which presumably are being sync'd with iCloud, will not be sync'd to your new Mac (and vice-versa).

If you are syncing your iPhone with your Mac via the Finder -

Connect your iPhone to your new Mac, open a Finder window. Then, in the left pane, select your iPhone, then in the main window headings click Photos, then UNcheck "Sync photos to your device from ..."