Yes, Shakespeare invented over 1700 of our common words. Please see my About, Inc. site for a list.

And yes, he invented many of the most used expressions in our language. Bernard Levin said it best in the following quote about Shakespeare's impact on our language:

If you cannot understand my argument, and declare "It's Greek to me", you are quoting Shakespeare; if you claim to be more sinned against than sinning, you are quoting Shakespeare; if you recall your salad days, you are quoting Shakespeare; if you act more in sorrow than in anger, if your wish is father to the thought, if your lost property has vanished into thin air, you are quoting Shakespeare; if you have ever refused to budge an inch or suffered from green-eyed jealousy, if you have played fast and loose, if you have been tongue-tied, a tower of strength, hoodwinked or in a pickle, if you have knitted your brows, made a virtue of necessity, insisted on fair play, slept not one wink, stood on ceremony, danced attendance (on your lord and master), laughed yourself into stitches, had short shrift, cold comfort or too much of a good thing, if you have seen better days or lived in a fool's paradise - why, be that as it may, the more fool you, for it is a foregone conclusion that you are (as good luck would have it) quoting Shakespeare; if you think it is early days and clear out bag and baggage, if you think it is high time and that that is the long and short of it, if you believe that the game is up and that truth will out even if it involves your own flesh and blood, if you lie low till the crack of doom because you suspect foul play, if you have your teeth set on edge (at one fell swoop) without rhyme or reason, then - to give the devil his due - if the truth were known (for surely you have a tongue in your head) you are quoting Shakespeare; even if you bid me good riddance and send me packing, if you wish I were dead as a door-nail, if you think I am an eyesore, a laughing stock, the devil incarnate, a stony-hearted villain, bloody-minded or a blinking idiot, then - by Jove! O Lord! Tut, tut! for goodness' sake! what the dickens! but me no buts - it is all one to me, for you are quoting Shakespeare. (Bernard Levin. From The Story of English. Robert McCrum, William Cran and Robert MacNeil. Viking: 1986).

For a comprehensive list of Shakespeare spinoffs, click here or visit the wonderful site, Titles from Shakespeare.

How did Shakespeare influence other writers?

Shakespeare influenced every generation of writers since his death and still today has some level of impact on contemporary plays, movies, and poems. So this is indeed a very large topic. If I were you I would select a few writers from each time period to reflect both how Shakespeare influenced them specifically as well as influenced notions of literature throughout the ages. Milton would be a good choice for the late 17th century, and you should read his well-known epitaph on Shakespeare, called "An Epitaph on the admirable Dramaticke Poet, W. Shakespeare", which actually appeared among the introductory notes of the Second Folio (1632). It is a wonderful testament to Shakespeare's influence on Milton, who, in this poem, discusses the immortality of Shakespeare's works. Milton also mentions Shakespeare in his text "L'Allegro." For the early 18th century I would cite Alexander Pope, who produced an edition of Shakespeare in 1725, with much commentary. If you browse through his collected works you will see many references to Shakespeare. In the 19th century (some might say late 18th century), I would highlight Keats as an example of how Shakespeare impacted later writers. Keats was so influenced by Shakespeare that he kept a bust of the Bard beside him while he wrote, hoping that the power of Shakespeare would guide him. Keats' poems duplicate Shakespeare's style, and are full of Shakespearean imagery. Moreover, in his personal letters to friends, Keats mentions several times his greatest role-model. In this letter to Benjamin Robert Haydon dated 10 May 1817, Keats writes:

I read and write about eight hours a day. There is an old saying 'well begun is half done' - 'tis a bad one. I would use instead, 'Not begun at all till half done;' so according to that I have not begun my Poem and consequently (a priori) can say nothing about it. Thank God! I do begin arduously where I leave off, notwithstanding occasional depressions; and I hope for the support of a High Power while I climb this little eminence, and especially in my Years of more momentous Labor. I remember your saying that you had notions of a good Genius presiding over you. I have of late had the same thought, for things which I do half at Random are afterwards confirmed by my judgment in a dozen features of Propriety. Is it too daring to fancy Shakespeare this Presider? When in the Isle of Wight I met with a Shakespeare in the Passage of the House at which I lodged - it comes nearer to my idea of him than any I have seen - I was but there a Week, yet the old woman made me take it with me though I went off in a hurry. Do you not think this is ominous of good? I am glad you say every man of great views is at times tormented as I am....

This King Shakespeare does he not shine, in crowned sovereignty, over us all, as the noblest, gentlest, yet strongest of rallying-signs; indestructible; really more valuable in that point of view than any other means or appliance whatsoever? We can fancy him as radiant aloft over all Nations of Englishmen, thousand years hence. From Paramatta, from New York, wheresoever, under what sort of Parish-Constable soever, English men and women are, they will say to one another, 'Yes, this Shakespeare is ours; we produced him, we speak and think by him; we are of one blood and kind with him. (Thomas Carlyle: The Hero as Poet 1841).