Generally, yes, it can. You can even find instructions about how to fix this problem.
More specifically, Chrome sync synchronizes (per the advanced sync settings of Chrome v.47):
- Apps and extensions
- Autofill information
- Open tabs
If any of those can be maliciously manipulated, it will likely be spread by Chrome sync. Here's malicious activities that I can think of; I'm sure there are more:
- Apps and extensions - Many, many ways to do bad stuff here. These can open windows, alter page contents, redirect pages, etc...
- Autofill information - An attacker could try to have online orders shipped to the their address or fill in other information.
- Bookmarks - Alter your bookmarks to go to attacker's site.
- History - As this is used for completion, you can misdirect the user when typing an URL to go to the attacker's URL.
- Passwords - Assuming you can only set passwords and not get them, all that I can think of is that an attacker could trick a user into logging into a site as the attacker. Presumably the user would enter confidential information or do something else that would be of value to the attacker.
- Settings - You mention proxies. Therer's also the settings for the new tab and home pages. I'm sure there's more.
- Open tabs - Open pages with malicious content or downloads on remote devices.
- Themes - Can't think of much. Perhaps change the user's theme to lots of cats?
Destroying data in Chrome sync may also be a way to spread malicious activity. For example, just deleting all of a user's bookmarks could cause great pain for bookmark-heavy users.
While every piece of browser malware may not be applicable to being synced across instances, certainly there are many ways to do it.