Browsers generally implement a level of isolation between the websites and the browser. So a website may not be able to directly fetch the browser history or cookies created by other sites. Services like Google, Facebook, Twitter etc are ubiquitous due to ads, like buttons etc.
It is possible for these services to fetch your browsing history by indirect ways. They won't be able to get all the cookies on your browser though as the cookies are bound to the url that created them.
What will happen if I login in Google Account and keep it logged in. Will it collect my browsing history and cookies?
Browsing history via browser? No. Google ads can create cookies saying you connected to this website that uses Google ad. This data can then be used by Google to build your browsing history
What will happen if I logged out after successful login in Google
Account. Will it be able to collect my browsing history and cookies,
or something else?
Logging out temporarily stops the Google service from being able to link the browsing to your account. When you log back in, Google can look at the cookies that it stored to get an idea about the browsing history on that macine and possibly link it to you. Of course, it won't always be able to link that browsing history specifically to you. For example, if you are sharing a browser with multiple people, Google may not know who was particularly browsing the computer if nobody is logged in.