I am a licensed audiologist in the US and currently practicing in Hong Kong. The general rule for the ear is NEVER STICK ANYTHING SMALLER THAN AN ELBOW IN THE EAR. I have seen horrible complication from earwax removal done by parents. For starter, an infant's ear canal is very narrow and if you put a q-tip in the ear canal, you end up pushing it further down, making it dry and pushes towards the eardrum. Earwax is a natural material that our body produce to prevent dirt and hair from getting to the bony portion of the ear canal (which you usually would not be able to reach since the ear canal is bendy and the bony part is past the second bend of the ear canal... however, if you do reach it, your child will jerk, since it's very delicate and sensitive).
Also, do not put any eyedrops in the ear canal either. What is doing is to mix the drop with the wax, making the wax to expand and becomes more mushy and does not help the natural wax to come out carrying the debris in the ear canal.
I have always told parents to NOT do anything at home. Also, do not allow your GP, any nurses or pediatricians to remove earwax for your child. The only person that should remove the wax is an Ear, Nose and Throat (ENT) specialist. They usually use a special air suction system to remove earwax, which is a lot safer than irrigation. In addition, they are trained to do so. The reason we do not recommend irrigation is that we use a water irrigation system to test the balancing system in adults (shooting water down the eardrum will activate the balancing system in the ear and makes the patient very dizzy). If the nurse accidently shoot the water down in the wrong angle, it will make the child very uncomfortable. ENTs are a little more expensive, but they are worth the money if you are concern about your child's earwax. If you find your child's earwax being very hard, we recommend using the dropper and put 2-3 drops of olive oil in the earcanal 3-5 days before you visit your ENT doctor. It helps soften up the earwax and makes the suction a lot more comfortable.
If you leave the earwax untreated or keep pushing it towards the eardrum, it will cause a CONDUCTIVE HEARING LOSS. Such hearing loss is usually temporary and hearing goes back to normal once removed. However, a conductive hearing loss can cause a maximum of 60 dB HL decrease in hearing, which is siginificant for a child who is at a critical age for speech and language development. So take your child to see an ENT and have him examine the ears. If the wax is completing occluding the eardrum, he would remove it for you. Otherwise, it's nothing to worry about.