This is a general tip that goes beyond pitch and intervals: Learn to sing things that you play and hear. This is good for a lot of things. It's a way of developing your ear without the mechanics of your instrument getting in the way when you're starting out. You can "practice" when you don't have access to your instrument, even listening to the radio in your car. It will develop recognition of familiar musical patterns.

You can practice singing the intervals. What's a fifth sound like? You should be able to sing it. Or play it on your instrument and then sing it.

Yes. Focus a few weeks on just learning the sound of a particular degree of the scale. Like you are just trying to teach what the dominant sounds like ('Ruffles and Flourishes' is appropriate for this, as an example). After it's correctly learned you go to a different one. After awhile you've taught them all.

Everyone can learn good relative pitch with practice. Music schools do this regularly, and it's just a skill you can pick up. Start by identifying intervals, then learn chords, and then learn to write down music you hear, and so on. It just takes work.

"Sight singing" is the classic exercise to develop strong relative pitch. There are lots of resources on this — there was even a sight singing class at my college. It might be a little too challenging (and boring) for a young kid though.

If you're willing to give the app a try, I bet it could actually be a pretty solid way to learn relative as well as absolute pitch. Just manually play "Red" before you start to anchor yourself. I've noticed some improvement in my relative pitch just by practicing it with my daughter. I'd be interested to know if anyone ends up using it explicitly for that purpose.