Three medium/large fish in a 10 galon tank could push the ammonia up skyhigh in a day or two. If, for some reason, the filter slowed or stopped filtering efficiantly, the beneficial bacteria would have less contact with the ammonia in the water, less chance to process it, and the ammonia could easy get out of hand. If, for some reason the oxygen available to the beneficial bacteria was reduced, the same thing could happen.
Keep the tank cool.
The Penguin 150 is a great filter for amounts of water going through - 150 in a 10 gallon tank should be ok, even for a dramatically overstocked tank. The problem is the cycling bacteria. If I remember correctly, that particular Penguin relys completely on the bio-wheel to hold your beneficial bacteria. When you change out the cartridge, you have thrown away anything that might be on that, leaving only the bio-wheel.
I like bio-wheels. I use them in almost all my tanks. They are a good place for the cycling bacteria to colonize. BUT....they are really not the best places. They have limited surface area - and can hold only so much bacteria. In a tank that is overstocked and needs the maximum amount of beneficial bacteria to process the excess ammonia, that wheel is, sadly, simply not up to the job.
There are a couple of things you can do.....
First, do not throw away that cartridge! You can either slit the top of the cartridge and bang out all the carbon chips into your garbage. Then, buy some ceramic rings, lava rock, sintered glass or other biomedia that is sold for filters. Poke that into the slit of the filter cartridge, filling up your cartridge with biomedia. The floss will still filter out the majority of the large waste, as it is meant to, but the bio-media will support a huge colony of beneficial bacteria - a nice backup to the bio-wheel. When you wish to clean this filter, just bang it around in a tub of old fish water, and/or, pour out the media, blast the floss clean in the sink and pour the media back in.
SEcond, if you want to make a "permanent" bio-media cartridge, You can make a netting bag or even use a stocking. Peel/cut all the blue floss off the cartridge frame and stick it in the netting bag. Fill it with bio-media.... This can fit behind the standard cartridge in the filter - it takes a bit of finagaling, but it fits - and will hold your bio-cycle no matter what happens to the wheel or the cartridge.
Upping the surface area will up your colony's population of beneficial bacteria. The more bacteria, exposed to the ammonia in the presence of sufficiant oxygen , will result in the maximum amount of ammonia to be processed.
If you have gravel in the tank, removing that will not affect your cycle (gravel holds little beneficial bacteria - only the very surface really holds much at all - so you will not lose too much), and having the gravel gone will give your fish all the more water volume to live in.
This, in combination with major and frequent water changes should allow you to keep those fish alive in a 10 until you are realy to upgraded.
If you have a 55 gallon tank, all you need it is a level, supportive surface that can support 450-500 pounds of weight. A stack of carefully placed cinder blocks will raise the tank up sufficiantly that your syphon should work well. A lid/hood, with the lid taped down should be fine. A 3 year old is too young to be running without supervision anyway, so there should not be too much problem. You could also use the basement, a spare room with a door, or even block the area off with carefully placed furniture, boxes or whatever.