Do background noises like chatter or phone alerts hurt our memory and focus?

It depends on the background noise. There have been studies that looked at a different number of voices in the background when people were doing a memory task. One or two voices were very disruptive, but once it became a larger number of voices it was not as disruptive. In essence, with a larger number of voices (like 16) the individual words/voices could no longer be distinguished and it became less disruptive because it was more "general noise" instead of specific conversations.

Phone alerts are designed to capture our attention and get us to switch away from our focal task and to attend to the phone. They are unpredictable, which makes them much more likely to distract a person away from whatever they were working on before the phone alert. If a phone chimed (say every 15 minutes) it would be predictable and much less disruptive.