Any tweet you send will be shown on your timeline and also on your profile.
To see who has tweeted you, click on your 'mentions' tab on your homepage. There, you will see every tweet that you have been mentioned in.
No they won't see your tweet.
Yes, when you tweet someone directly by mentioning their handle (using @username), they receive a notification about the tweet. However, if you tweet something without mentioning them, they will not get a direct message or notification unless they see it on their timeline. Additionally, if their account is private, only their followers can see the tweet if they are not mentioned.
Yes
Yes. As long as you don't start the tweet with the persons username and mention them in the body of the tweet.
It is normal to not see your tweet on the Twitter timeline because sometimes it takes several minutes for it to show up.
Yes, a person can see your tweet, even if it's a day old. As long as the tweet hasn't been deleted, the other user can view it.
Depends on what you're talking about. If you start a tweet with @username, only that person you mention will see it. Now, if you place @username anywhere else in the tweet, everyone who follows you will see that tweet in their timeline.
YES.
As long as you include their username in your tweet (@username) then your tweet will appear in their @replies tab.
yes. you only see people you follow.
Yes you can still tweet them and they will still see the tweet in their mentions.