When you fill out the E Ticket and the Code comes there is actually a click you can make to Email it to yourself. Use it and you can also print it. But having it on your phone is all you need........
Airlines ask to see the code, just to make sure you have gone through the process. I think the airlines submit the manifest (passenger list) to DR immigration and they tie the passenger passport number, name, date of birth to the e-ticket record to create a full data record. Someone told me that's how it works anyway.
Remember if you are staying in Santo Domingo, areas like the Colonial Zone, Naco, Piantini (the usual tourist areas) you DON'T pick Santo Domingo from the list of areas, you pick Distrito Nacional, to enter the address. And if you're doing it for two people, it asks right on the first page if you're travelling with anyone else, look out for that it's quite hidden.