What stops Google from saving all the information on my computer through Google Chrome?
What processes and systems are in place so that Google is not able to copy the data on my computer?
None. Google Chrome usually runs with the permissions of your user account. The application can then read and modify local files to the same extent your user account can. (These permissions apply to most of the programs you're using.) So you need to trust Google in that they don't ship a malicious update that spies on you, or keep sensitive files inaccessible to the account you're running the browser with. Alternatively, there are most likely sandbox implementations for your OS that let you run Chrome in an isolated environment with restricted access to the filesystem.
What processes and systems are in place so that someone who writes a Chrome extension is not able to copy files on my computer?
Chrome extensions have limited privileges by default. An extension needs to explicitly request (declare) a permission to interact with documents on the file:// scheme.
Also note that your browser disallows ordinary websites to read or even redirect to file:// URIs. So while your local files are accessible to the Chrome process, they are not exposed to the web.
A piece of unsandboxed software running on a PC/Mac has (generally) the same privileges as the user running it and therefore can access any data that can be accessed by the user.
You are trusting Google (and any other software vendor whose code you execute) not to do anything malicious with that access.
If you don't trust Google, your only option as a general user of software is not to run their code.
The situation with Chrome extensions in somewhat different. Google places restrictions on what the extension vendor is able to do, and when you install an extension it will tell you what permissions you are providing it.
Obviously you are still trusting Google to have coded these restrictions correctly, and you are still potentially trusting the extension authors with some permissions that could be used to take malicious actions.
If you are running a Linux distribution with SELinux, it is possible to have an additional layer of security. SELinux is an OS-level technology which allows tight restrictions on what processes — like your browser process — can access. In fact, in Fedora and in Red Hat Enterprise Linux (disclaimer: I work for Red Hat, on Fedora!), there is a light confinement by default for Firefox and Chrome. This turns out to be difficult and inconvenient to make more strict for most users — see this blog post by SELinux guru Dan Walsh for more.
There is ongoing work in Linux in general to run more user-level applications with greater restrictions (see for example Flatpak).