Some things to think about for future cakes ... great entries in the cake diary that others have mentioned .....
The "small square" that your 2-year old decided to sample ... was this a square cake? a square gumpaste plaque? a square what? If it was a cake, why was it in the 'frig? Unless you are using a perishable filling (and if you're using canned icing, I would presume any fillings would be shelf-stable, also), there is no need to refrigerate a cake.
You started with one round layer ... so this was going to be a single layer cake? Then you thought you wouldn't have enough cake? So decided to make it a 2-layer cake? Here's a serving chart you need to print out and tape to the inside of your cabinet door: http://www.wilton.com/wedding/wedding-cakes/wedding-cake-data.cfm
This chart assumes a 2-layer, 4" tall cake. This will help you know up front how big a cake you will need and how many cake mixes/batter you will need.
Make a list of how many cake mixes it takes for each pan so you'll always have the right number of mixes handy.
If the 2nd layer you baked wasn't the same height as the first layer, it's not a total loss. Once you ice it, no one can tell. And once it's cut, no one cares.
The cake doesn't have to be cooled before you level it. I level my cakes while they're still in the pan and my cakes are leveled and flipped out onto a cooling rack within 2 minutes of coming out of the oven. Sounds like you need a sharper or better quality knife.
I've no idea how big the cake was that you tried to flip, but use a cardboard and slide the cake onto the bottom layer rather than flipping it. Much safer. As you found out, you can't "scoot" a cake across icing. Use your spatula to lift and move the cake. Scooting will definitely cause a problem. Unless the cake is partially frozen, then you can handle it a bit easier, but a partially frozen cake can also be lifted and moved real easy.
Find a good icing recipe and make your own. (1) canned icing is designed to be spread, not really designed for decorating. (2) if you run short, it's easy to whip up another batch instead of taking the time to run to the store.
As mentioned above, a paying client is not who you should be practicing on. I'm not trying to be harsh, but the issues you ran into are pretty basic and you should be past that level before accepting money for a cake. I'm sure some will disagree with me and that's ok .... all of this is offered in the spirit of helping you master those skills and get to the point where lots of people want to pay you for a cake. If your goal is to do this as a business, then you have to be serious and get serious about it. That means investing the time and commitment to the task, to the skill, to the vocation.
As cabecakes suggested, a wilton class will get you thru the basics and get you going in the right direction.