In answer to the first question - yes, it is possible in a pragmatic sense, and running a separate network will not appreciably impede functionality, assuming the separate network is given sufficient access to the Internet to allow basic connectivity from the device(s) in question to its / their respective "home base" services.
In answer to the second question - while it is advantageous from a security perspective, I would not deem it to be "significantly" advantageous. Many of the current exploits around IoT devices involve vulnerabilities on the service side, not necessarily on the device itself. Separating the networks between "regular computing" and "IoT" in your home will help mitigate risks to your home computing infrastructure, but will not reduce risks to the devices themselves, or to the Internet in general from devices that have been taken over via vulnerabilities in the services they connect to.
To put it differently - if a bad actor takes over the Nest server network, that actor will be able to access your Nest thermostat whether it is on a separate IoT-only network at your home or on a common network with other compute devices. The separate network will only help protect your home computers in that scenario, it won't help protect anything else.
As to the follow-up question about consumer devices such as phones - that decision will have to be made on a case-by-case basis for your personal use cases, but I would wager to say that most of those devices would end up on the regular compute network, especially if you will be syncing data to/from your mobile device and your computer directly (as opposed to using a cloud-based sync like Google Play or Dropbox).