I've had a look at some of these. Looks really great for kids (and any beginner, but was designed for kids). Frankly I'm jealous of anyone who would get to start out like this.
https://www.newinchess.com/The_Complete_Step_by_Step_Method-p-1933.html
How much improvement is possible? Kids can improve a few hundred points a year, every year... but 1st grade is what, 6 years old? That's still pretty young. As long as he's enjoying it and learning things that's fine even if the rating is about the same (yes you can learn and still not preform better). They can suddenly gain 500 points in a single year when something clicks and they start putting it into practice. If he were 10, working on chess, and not improving, then something needs to change.
As for USCF, I've played casual games with kids who love the game and work really hard... for about 20 minutes, then they get bored and start to play really poorly. At that age I think it's more about focus and temperament (after a mistake do they get really upset for example). So it's hard to say.
My son is entering 1st grade. He played a couple unrated tournaments in NYC last spring and liked them, then did a a lot of chesskid 15 | 0 games this summer. He he currently rated 1015 on ChessKid fast.
1. What's the best way for him to keep improving?
2. What do you think this translates to in USCF rating?
3. How much improvement would be possible in a year if he works pretty consistently?
TIA!