No, you don’t need a Swiss credit card or a Swiss phone number to register.
It is a wrong conclusion to assume single, individual tickets are cheaper than day passes. From a cost perspective, they are identical. When it comes to fare network tickets, and ZVV is the fare network for the greater Zurich area, return tickets automatically turn into a day pass. A ZVV day pass is valid for a duration of 24 hours, and any other day pass is valid from today at 00:00 to tomorrow at 05:00 AM.
You are welcome to use the ZVV app. Using the ZVV app (or equivalent) requires good knowledge of the ZVV zone map and the routes serviced. I recommend purchasing tickets at the counter if you are uncertain about which ticket to purchase. If you are caught with a wrong ticket, or if you are missing the required zones, you will have to pay a surcharge of CHF 100 - that would be a pity.
If you travel throughout the country, you may wish to purchase Saver Day Passes, which are also valid for local transport. Furthermore, you can use the SBB Mobile app, which is the official app for purchasing public transportation tickets. Also, depending on your overall itinerary, you may wish to buy the Swiss Half Fare Card, it can save you money, too - especially if you plan for mountain trips.
Saver Day Pass – book earlier, travel cheaper | SBB
Swiss Half Fare Card – Switzerland at half the price | SBB