1. as far as I know you don't need to be signed in your G-Account on Chrome to use Google services on the web.
2. there's an option in Chrome to never store your history (called incognito mode)
3. With google play you mean google web store, where you can find apps for Chrome?
4 "I guess that it designated all of the older stuff on the browser as done by me, and now if I log out of chrome it gives me a blank-slate browser." not really sure what you are trying to say here.. can you explain it better please?
tbh using logging in with your G account in Chrome it syncs bookmarks, preferences etc, in the cloud, so good if you have more than one machine (including phone)
imho the price that I pay for Google Services (I give them my data) is more than fair for the quality and versatility of these services. they are not interested in your particular data, but in the so called "big data". they dont care if john smith is going on vacation next month, but how many people are going to vacation in (ie) august. you are just part of a big pool.
use internet for legal stuff (obvs don't buy drugs online, that's just stupid) and you'll be fine