I dont think a pediatrist is necessary. If anything, it would be beneficial to see a physical therapist or athletic trainer.
You should never limp when walking, it just creates poor movement patterns and will create problems in other areas (hip for example). It does sound like a basic ankle sprain with a probably peroneal strain.
Being 1 week out of the injury, I really don't think excessive ice is necessary. If you want to do it to numb pain after activity, go for it, but it doesn't need to be as frequent as when the injury initially occurred. Focus on ROM of the ankle and strengthening of the ankle. Spell the alphabet with your foot, make big circles, etc. If you are able to get formal therapy, see if the therapist is familiar with ankle traction techniques. I have found those to be very beneficial with ankle sprains. Seeing as it seems like a mild ankle sprain from the info you've given, you can even probably begin some balance/proprioceptive work for your ankle.
1) Stop limping
2) Ice after activity as you see fit - not necessary to ice all the time though this far into the healing process
3) Work on improving ROM and strength of ankle
4) Begin basic balance/proprioceptive work