So I just went and watched that specific scene because you do raise some good points here.

If the computers know he woke up why release him instead of just killing him?

It looks like the machine that releases him is a different one from the usual Sentinels we see elsewhere in the films so I assume it's just there to monitor the humans in the pods. It is probably programmed to just dispose of humans into the sewer that either wake up due to technical issues or that die in the pod by yanking their neck using the grabby arm and unscrewing the neural spike. Flushing the live/dead body down is just the last step in it's process because even if they wake up alive in the pod, once they're flushed down they aren't going to be fed nutrients to keep them alive and these vat grown humans barely use their muscles so they can't swim well and will most likely drown quickly. Why bother wasting energy making sure they're dead when they most likely will die anyway?

If they did know he did woke up, and didn't kill him because he's the one... Wouldn't the crew find it unusual how easy it was to get him?

I don't think the machines knew he was The One at that stage because he was yet to perform any of the feats that the "prophecy" laid out. I don't think I could explain the whole prophecy thing they go over in the third film well enough but from the sounds of it, until Neo can see the Matrix for what it truly is the machines don't know if he's the one or not so until he gets shit by Agent Smith and comes back to life to the machines he's just another human that broke free and being a bit of a pest like the rest of them.