Using Android without Google account
Of course you can, in fact, the only reason a lot of people connect their Google account is to use play store and get apps easily. 

I suggest you to install F-Droid: it is the best source for open source Android apps, where you can find a "copy" of all g-apps. 

For the last part, technically, if you answered No to all google's attempt to get you accept their therms and conditions, they can't. But keep in mind that a lot of applications are based on Google API, so, they would use things like Google Play Services to get some jobs done. (I'm not sure, but I believe they pass all information directly to the app without saving anything on their databases about you if you did not accept TOC)

If you really want to avoid Google, I suggest the use of a custom ROM