free 3 month offer on my Roku?

When you redeem an offer for a 3 months free trial to Apple TV+, then you will start a subscription on your Apple ID account that is valid on any of your devices that can bring this Apple TV+ content, including through the Apple TV app on Roku.

Redeem such an offer on the device that made you eligible for that offer. Follow the redeem instructions as indicated for that offer. Apple has offers for an extended trial, in combination with the purchase of select new Apple devices. Select partnering companies may also have offers for Apple TV+, in combination with purchases of select products or services, with free trials of various durations, and with their own terms and conditions, and with their own redeem procedures. What offer did you get, and through whom?

Yes, that will work. Sign in to the Apple TV app on Roku, through the in-app Settings (sprockets wheel icon), with the Apple ID account that has the subscription.

List of Apple TV app-supported smart TVs and third party streaming devices - Apple

Get the Apple TV app or Apple TV+ app on your smart TV or streaming device - Apple Support

How to watch Apple TV+ and more - Apple Support

↳ Smart TVs and streaming devices﹥Roku﹥

Apple TV app user guide for smart TVs and third party streaming devices - Apple Support