@Cshueh
Two of our Google Nest Hello Doorbells have been peeling for some time. We bought silicone skins on Amazon to cover the peeling shards, since the doorbells are beyond the 1-year warranty.
Regarding the second problem, it sounds like your doorbell's internal battery (yes, it has one) has failed. This causes the doorbell to go offline for a minute whenever the doorbell button is pressed, which also means it doesn't ring the internal chime or send a notification.
This has now happened to all three of our Google Nest Hello Doorbells, and has affected hundreds of other customers (https://www.googlenestcommunity.com/t5/Cameras-and-Doorbells/Nest-Hello-Doorbell-cuts-out-when-butto...). Since Google Nest does not replace such doorbells if they are past the one-year warranty, we've turned off the "Indoor chime" option. This prevents the doorbell from going offline, since the purpose of the internal battery is to provide power for the doorbell during the instant when the doorbell button is pressed and power is diverted to the doorbell chime. We instead turned on the "Visitor announcements" option (https://support.google.com/googlenest/answer/7672451) in order to get notifications instead on our Nest Hubs and Nest Mini.
Some customers have opted to replace the internal battery, but this involves tearing apart your doorbell, and it doesn't always work:
https://www.ifixit.com/Guide/Nest+Hello+Battery+Replacement/131729
