You can see how many tweets you have posted by going to your profile. There is a total number underneath your number of followers/people you follow.
If your tweets are protected then the "retweet" button is not available on your tweets. However, this doesn't stop someone who is following you from copy and pasting your tweet to a new tweet and adding "RT" before it.
The plural form of tweet is tweets.
You can send a maximum of 60,000 tweets per day.
No you can't.
At the top of the homepage, it will regularly display how many tweets there has been since your last refresh. You can also see tweets that you have been mentioned in by clicking on the @username tab on the right hand side sidebar.
You can tweet about anything, there aren't any "set" tweets that people use.
No, they can't tweet you.
The only way that your tweets will appear on their timeline is if they are following you. If you want someone to see a tweet but they aren't following you then you should 'mention' them in your tweet (@username)
The only thing that changes when you change your Twitter name is the name at the top of the tweet. The content of your tweets and the number of retweets/favorites stays the same.
As long as you include their username in your tweet (@username) then your tweet will appear in their @replies tab.
The limit is 100 but, after that you can't tweet for like 10 mins then you can keep tweeting
They won't appear there if you don't include their username in your tweet.