Ok
Here is how I would approach things. You want to get the best value for your money, so some careful shopping is needed.
Used gear represents the best value, but there is risk that it could break soon, or need service. If it is at all possible see if you can buy a used piece that has been restored or from a store that puts a warrantee on it. It costs a bit more to do this way, but there is a lower threshold of risk. One place that does this is echo hi fi in Portland OR, and they ship. There may be other places closer to you in one of the bigger cities. This is what I would do for the integrated amp or receiver. An added bonus is if you get a unit from the 70's, 80's, or maybe the 90's you often get a quality built in phono stage.
I would find a good cheap used DVD player for your CD playing. Or buy an inexpensive USB DAC and use a lap top as a music server. Not all DVD players sound the same, some are better than others. I would look for a Phillips or a Sony one. You should be able to buy one for dirt cheap, less than $15.00 really.
I would opt for smaller floor standing speakers and use the money spent for stands to get a bit better speaker. Speakers and stands are a dangerous combination with kids, and a small floor stander is not immune to being knocked over, but it is less likely to happen. Also you will get a bit more bass out of them due to larger cabinet size, and so you might like them a bit better. Also I would look at discounted sites on the web for speakers, and if you are really lucky you can get one that offers free shipping. Now there are good options on the used sites also. If you can find some that ship for a reasonable price, then look at the Canadian offerings from companies like Paradigm, Angstrom, Energy, Mirage, and what not, there is a lot of value there. It's usually the shipping costs that make used speakers a bit more problematic to find on a tight budget.
One other option to consider is joining AK Bartertown, which is where members buy and sell to each other. You may very well find a good quality used turntable here, from people who know how to care for and pack and ship one. Heck you could find your CD player, integrated amp, even your speakers here. It could be one stop shopping. Is iron clad proof that you won't get a bad transaction? No. But you are dealing with a good bunch of people, and paying used prices from hobbyists.
As far as the Amazon card goes. Can you buy general household needed items with it, and set the cash aside that would be spent for the audio purchase? It would give you the flexibility of having more vendors/places to shop at. And it doesn't cost your family any money out of pocket, although its not quite as convenient, and costs you extra work.
Just some thoughts from a simple pig. We like god food, warm sun, and soft hay. If you want a different opinion, go talk to the cows for they are wise. Don't talk to the chickens, fowl are known liars and they cheat at cards.
Regards
Mister Pig