I think you will be fine with your price range. If you really just want to tool around with your kid now and for the next few years and nothing more, you will likely be fine with your choices. If you will do some riding outside of your kid, I'd recommend spending a little more time picking your bike. Visit a few local shops over the next month and test ride a bunch of bikes. Over time you will start to have some favorites. Based on the price ranges of the bikes you mention, there likely are differences in the bikes that may not be readily apparent to you at this point. It's very hard to help with the nuiances of what bike people would recommend without more conversation. I just want to help you understand what you could expect on the forum. Try searching for other threads on this issue. There should be some. Many have talked about the Trek 7.3 or 7300 favorably, but that is a comfort bike. However, those threads may discuss some of the finer points of opinion in bike selection.
If the bikes feel about the same handling wise, consider the weight and the components of the bikes. If it is significant, I would go for the lighter one with the better components, all other things being equal. There are some caveats around that statement, but close enough for your purposes without boring you
. I am not a fan of aluminum but that is my personal preference. To me, they feel much harsher than steel or other materials. They are/were wonderfully light. Many people love their aluminum frames. It would be better for you if you did like aluminum as you would have that many more choices. Component levels should come into play also. Better components will last longer and perform better. They will cost you more upfront but should provide better service (very analagous to computers and tools in terms of tradeoffs). I do not know any of these bikes so I have no specific comments for you.
F300 reviews - http://www.mtbr.com/reviews/2003_har...t_121460.shtml
You will need specific model numbers to hopefully find other reviews on mtbr.com.
One option is to get the less expensive bike now until you figure out what you want. You can always sell the bike and get something you like more later.
a) It typically does not matter where the bike is manufactured. In your price range, they will mostly be asian manufactured.
b) For @$500 they are more or less the same. However it was my opinion that you could get more in a MTB bike for a given price than you could in road bikes, in the mid-price and up range. That may no longer be true as I do not currently keep up with bike prices.
c) If you are going to do some mountain biking, then you will want the better performing components. Mountain is taxing on the components and they need to perform quickly and reliably to get you through technical single-track riding. If you're talking fire roads, not as much of an issue.
Whatever you decide, don't let Spring go by without you purchasing a bike. Your kid will never be 3 again.