It changes over time, sometimes there will be a dismissable popup, for a while it couldn't be dismissed. It does seem to work now for non-nsfw tweets, but nsfw login-walled ones are indexed by Google too I believe.
Specific tweets, yes, but if you want to scroll a person's profile, you'll need an account after about a page of scrolling.
no, just close that popup? if it even appears... when is the last time you've used twitter
Logged out - about two weeks ago, couldn't close the pop-up, no X button. Perhaps it was changed recently but doing it on mobile was simply not an option for a long time.