No more graphics card with 2nd generation Intel Core processors (i7,i5,i3)?

The Sandy Bridge line of processors each include some version of Intels HD Graphics - either the 2000 of 3000 series chips. In order to use this feature, you must use either an H67 series board or a compatible Z68 board. Note that H67 boards lock overclocking features, but Z68 boards leave it available.

To answer your next question, the processor you probably want in your desktop is the 2500K. The 2500K includes Intel 3000HD graphics and overclocks to insane levels. Even if you don't want to overclock, the little extra cash you dole out will go toward the ability to use a slightly better graphics card.

If you find it's not quite enough, you can always add a separate card and benefit from the speed of that card.

It's true. All of those processors have a graphics core integrated into them. They'll get you through photos and video viewing, but if you want to play any video games then you're going to need a discrete graphics card as well.

There hasn't been a need for a graphics card on most motherboards for the last 5 years. That doesn't mean you won't need to install a card if you want to do some serious gaming or autocad work.

Only with certain motherboards that have the H67 and Z68 chipset.

Anyways, I think you are referring to integrated graphics which has been around for over a decade now!

Many motherboards come with integrated chipsets -- meaning that a dedicated card is not necessary.

It has never been necessary for a discrete graphics component to be installed on a computer. All computers have integrated graphics.