I don't think you need to pair the phone to the computer. I've just tried it with my phone and my partner's mac. Don't complicate things by having iTunes open or it will start asking you to sync with the new phone - so close iTunes.
Using Photos - Connect the phone and Photos will probably open automatically. You will be asked on the phone whether you want to trust the Mac, say yes. In Photos, in the menu pane on the left, under import, click on the phone and all the iPhone pictures should begin to show as thumbnails on the mac. Click the ones you want to import, or CMD A to select them all, or select 'import all new' at the top, then import. There is a tick box if you want to delete them all once imported but I am always a bit worried I will delete them from the device but not actually import them, so I usually import then go back and manually delete the pics from the phone once I am happy I have them all saved.
For airdrop - make sure airdrop is enabled on your phone - swipe up from the bottom of the screen to open control centre, then tap on airdrop, and select everyone. Enable it on your mac - open a finder window, click airdrop, then set 'enable me to be discovered' to everyone. Go to photos on your phone, select a photo, then hit the share icon (bottom left with an arrow coming out of a box) and if your mac is nearby and unlocked, you should see it in the airdrop pane. Once you have a connection, you can tap all of the photos you want to airdrop, and send them all at once. By default they will go into your downloads folder.