Haha! That is one of the most controversial topics in the world of bird keeping. It's really up to the owner. I've seen huge changes to the birds confidence one it has a mirror. Some people think it's cruel.Quite the opposite! If you have just one budgie, and he has a mirror, it will drive him crazy and cause behavioral problems because the mirror will not interact with him like another live bird would. It will frustrate him, or at the least, cause him to develop an unhealthy obsession with the mirror-bird.
If you want your budgie to bond with you, don’t give him a friend, whether live or mirror-friend. You be his friend. It’s ok if you’re gone at school during the day. Give him plenty of attention before and after, and he will learn the routine. Daytime while you are away will become his nap time.
Also. As to the cage. What I meant by 'small' was NOT one of those tiny little cages you can buy from the pet store. A cockatiel size cage would be fine. You'll have to check the width between the bars as to make sure they can't escape though.
Get something like this-
As opposed to this-