Good first draft, Shaffy. But note that you may well have to rewrite it based on comments here.
I'm going to offer some ideas, but it's up to you to incorporate them.
First off, add an extra line break between paragraphs. It makes it much easier to read.
The Big Bang, despite the name, was anything but an explosion. It was an expansion of very dense and hot matter that had previously been compacted closely together. Much like gas held under pressure in a tank. That also gives you an easy way to illustrate why the universe has been cooling off in general - because as a gas expands, it cools. You can demonstrate this by taking a can of compressed air and spraying some air out of the nozzle, causing the can to get cold.
Inflation and the stretching of space are good as far as I can tell. There's no boundary to the universe as far as anyone can tell, but a good analogy is if you put a bunch of dots on a balloon, and then blow up the balloon. The dots stay at the same place, but the space between them increases.
However, it is not correct to say that objects in the universe have some velocity imparted to them by the Big Bang. It is the space between things which is expanding, thus giving the appearance that everything is moving away from us. Note my example of dots on a balloon - the dots themselves aren't really moving at all. It's the material of the balloon that's moving and stretching. The effects of the expansion of space are cumulative - meaning that as space stretches, the 'new' space also starts to stretch. This is why objects that are further away from us seem to be moving faster.
You also might want to note that we can't see past about 380,000 years after the Big Bang because before then, the universe was not transparent to radiation, because electrons and protons had not cooled sufficiently to form atoms until then. That is what the cosmic microwave background radiation is a remnant of - the time before atoms existed.