In most cases, yes, in all cases, no. There's a lot of variants of "enumerate all devices on a private network if you visit a malicious webpage" exploits.
Connections are only part of the danger. Browsers need to access the internet and they can have security bugs. Also, people download files and the programs that process them have bugs. People download programs and those can be dangerous.
For the kernel, it is important that it protect against dangerous programs running on the system to keep them isolated.
