What information or data concerning the activity of the Virtual Machine could be logged either by the Windows Machine (which the VM is running on)
Everything. The host has complete, unrestricted access to the guest. It may be possible in the near future to make VMs a bit more secure from their hosts by utilizing SGX (Software Guard eXtensions) on Intel Skylake processors. These use the CPU's Management Engine to encrypt data in memory so even the host's ring 0, the supervisor mode, cannot view it.
If you are asking what is unintentionally left over on the host, as opposed to maliciously snatched by the host, then it's mostly limited to swap (called virtual memory on windows) which is written to the disk when memory pressure is high, coredumps, which are performed when certain applications crash, and screenshots, which many virtual machine managers automatically take.
or by the network surrounding this Windows Machine?
Other than side channel metadata such as packet sizes and timings, an external network may be able to tell the guest's operating system version and uptime. See http://lcamtuf.coredump.cx/p0f3/ for an example of this, called TCP fingerprinting.
Note that a VPN does not obfuscate web traffic, it encrypts the contents of web traffic. This is fine if you are trying to hide the literal contents of your communications from your ISP, but it is not particularly effective at keeping you anonymous.