Is this normal for a 2 month old baby?

my daughter would be 2 months on july 15th and she has all the energy in the world! i mean she's up for almost the wholee day and part of the night and she wants to play and stuff, nuttin puts her to sleep. not her swing or pacifier or a bottle or anything! is that normal? the doctor said she's also developing fast bc she cud support her neck wen she was like 3 weeks and she was crawling at 5 weeks and she wants to stand and walk since 5 weeks too! of course i dunt let her stand nd stuff with all her weight on her legs bc its not good and anytime i see her crawling i stop her bc i think its too much for her legs...is all of that normal? this is my first child so i want her to take her time and grow up but it seems like shes growing up wayyy to fast :(