Yes, one betta in a heavily planted tank should mean no ammonia build up, and with no ammonia you should not see nitrite either. Just keep an eye on those two daily till you are sure.
As Matt said, 90% water changes (ie leave just enough water so the fish is still upright) are better than letting fish stay in water with ammonia and/or nitrite; the only thing to be careful of with such a large amount is that the new water is the same temperature as the tank water.
When you go betta shopping take your time. Let the betta choose you, don't go in saying I want a red half moon and nothing else. Avoid bettas that have finrot or fungus on their bodies, swollen eyes, scales like pine cones no matter how pretty they are. When you've had more experience you may be able to nurse such fish back to health, but if your daughter is very young having a sick fish die could be upsetting. Look for a fish that is alert; if they are in a large tank, look for one that is swimming round and not lying on the bottom. If the shop keeps them in small tanks next to each other, flaring at a neighbour is a good sign. Hold your finger next to the tank (don't tap on the glass) and move your finger round slowly - does he follow your finger, or even flare at it? That's a good sign. And finally, avoid a fish that is alert and active but seems jerky in its movements.
That's how I choose my bettas

Last September after my 3 year old betta died, I quickly narrowed it down to 4 fish, then it took ages to decide between them.
Feeding a betta - bettas are gluttons. Given the chance they will over-stuff themselves which isn't good. Once settled in they will beg for food - don't give in! And make sure your daughter understands this too.
They are best fed on specific betta food, with floating pellets being better than flakes. The brand most commonly available in shops is Hikari, but Atisons pellets - from Ebay or Amazon - are a better quality food. There are other good makes but they are sold in huge tubs which is fine for a betta breeder but not someone who has just one fish. Don't feed as much as the pack says, they want you to buy more.
Some bettas won't eat when you first get them. They don't recognise what you are giving them is food. The first time you feed him, give him one pellet/flake and see if he eats that before giving him any more. If he won't eat, remove the food and try again at the next meal. He will eventually eat when he is hungry enough, they have not been known to starve themselves to death.
You can feed 'live' food as a treat, bettas love bloodworms. A pack of alive food will go off before you've used much. Frozen cubes can be kept in the freezer for ages, but you need just a tiny bit of a cube per meal. Defrost before use. Dried 'live' food must be rehydrated before feeding. Those tubes of bloodworms in gel will also go off before the betta can finish one. Stick to frozen or dried. Bettas and bloodworms are like children and burgers - they'd eat them every meal from choice but they don't have much nutritional value, and are fatty. Once a week at most. Mine gets frozen bloodworms after his weekly water change.