Is it necessary to download an app to get my beats paired to my phone?

No. Beats headphones can pair directly to an iPhone. There is no Beats app for iPhone. You only need an app if pairing to an Android device.

Steps may vary depending on exact model of Beats Headphones.

click here ➜ Set up and use your Solo3 Wireless headphones - Apple Support

click here ➜ Set up and use your BeatsX earphones - Apple Support

click here ➜ Pair your Powerbeats2 Wireless earphones with your iPhone, iPad, iPod touch, Apple Watch, or Mac - Apple Support

Probably not. But what does the Beats instruction manual say?

It should not be necessary. Here is how to troubleshoot pairing Beats→If you can't connect to your wireless Beats product - Apple Support