The age-old question of fish and frogs, lol. It's a hard one to answer. . .
I've never kept Betta with frogs, only with other fish, but what I have learned with my four, and from what I have read - the best advice to give you is HAVE A BACKUP PLAN!
I've seen so many stories where it's gone swimmingly with the two (pun intended!), and also seen my fair share of horror stories of fish or frogs not getting along at all.
I think in a 10g tank, if you have it set up so that both animals will be comfortable, you stand a fair chance - but bear in mind how unique the temperaments of Betta fish can be. Each of my four frogs have been distinct individuals, too! Some seem more inclined to stand up and fight for their dinner, others tend to hide away. . .
I personally would be very comfortable keeping two frogs and a Betta in a 10g tank - as far as bio-load and tank size is concerned, but I also run fairly well-planted tanks, and that can have an effect on parameters - I've never done it the other way!
As for feeding. . . again! No easy answer. . .
My frogs never took to the dried food, and I've many read instances where this has been the case. But also have seen more than my fair share of pellet eaters out there. Personally, I would check with the shop and see what they're feeding them there. When I got my first frog, they told me to use the pellets, and since she wouldn't eat them, we were both unhappy until I figured out that they had been getting brine shrimp in the store. If they're already used to accepting the pellets, you shouldn't have any problems.
My frogs also love frozen Tilapia - variety in their diet is best, but I'm betting your parent's will be more comfortable with this being in the freezer, as it's people food >.< I'm sure there are other types of fish or shrimp that you could use to supplement their diet that come from the grocery aisle - something to look into, maybe?!
Freeze dried bloodworms are also an option that has worked for others, but that I have never tried personally. You'll want to be sure to rehydrate them in tank water before feeding to prevent possible intestinal issues.
Feeding fish with frogs without over-feeding or under-feeding. . . is a trick! But it can be done. You just have to pay close attention. My frogs have all be very easily trained to feed from my hand or a baster - and even from a specific location in the tank. Betta fish are good at learning routines, too. . . but you'll have to be wary and make sure that everyone is getting their share, and not too much. Adult frogs can skip a day or two between meals, too - so this can work in your favor.
Again - just be sure you have something planned in case it doesn't work out between them, so that neither frog nor fish comes to harm. . . and I say it's worth a shot. Please continue to do your own research, as well as watch this thread for other thoughts - I only know my own experiences, and what I've seen and read around the WWW!
Hope this helps! And please be sure to let us know how you decide!