Zurich is a very save city, I have worked and lived there at different times over the past 30 years and security would be the last thing on my mind, except for officially sanctioned day on anarchy - 1st of May.
Noise is always going to be an issue in any city, but in Zurich it is not too bad most have tribble glazed windows etc... the noise level is kept to a minimum. Quietness is a big thing in Switzerland, in apartment blocks it is unacceptable to flush toilets, have showers etc. between 10.00 pm and 6:00 am, so in general people tend to be careful about making noise late at night.
As far as I can see on booking.com there are several options well below your budget, but they are going fast.