I am looking to be in Beijing between the 1st & 5th of october will the zoo and underground be open

Yes, they are open at that time.

It's going to be crowded everywhere you go and there will be lines for everything you want to do as those dates are during the official Chinese National Day holiday. From my experience, it's one of the worst times to come to Beijing, unless of course you enjoy being jostled and shoved and crammed wherever you go.