What color for a modern area rug should we get for our living room?

We want a big one, 3 meters x 3 meters.

I'd choose an area rug that will pick up on the colors in your room without dedicating itself to just one color. An area rug such as this Beige Circles area rug would be perfect with your wall color and couch color. It would also pick up on the white of your wall unit.

When placing a large area rug, be sure to work it in as part of your room. Place it at the base of your couch or in front of your wall unit, depending on the location. An area rug can be a focal point, so make the most of it.

When you have a contemporary area rug, you can take shapes or colors from it and work them in as accessories in your room. With circles, you can add a circular vase, lamps, or photo frames. 

Is it ok to match black and white furniture?

Yes, you can match the two up by adding another strong color to your room as a feature color.

With a white desk and vanity, you can add a feature color such as deep red, hot pink, or turquoise. The color should be bright to offset the dark of the black.

Once you have your featured color, tie everything together with varied accessories.  If you have a solid white duvet cover on your bed, add a set of deep red or red/black blend pillows. On your floor, add a large area rug that features your colors.  A great example is a rug such as this Allure Rich Red Area Rug or this Beige and Blue Area Rug.  Be sure it's a large, eye catching area rug, and make it a focal point of your room.

Black and white furniture really goes well together if you do it correctly. Have fun with it! 

What is a good wall color for a living room with a white tile floor?

With a floor color like white, you can do almost any color. I'd focus on dark colors though, for at least one feature wall at least. Choose a chocolate brown on one wall and then a slightly lighter color on the other walls. You can put a great piece of artwork or family photos on the dark wall, and it will stand out against the floor.

Warm up your tile by adding area rugs. A warm, cozy shag would look great against white tile. Put something like this Cocoa Flokati on your floor. It  would look great with chocolate/taupe walls and dark brown wood coffee tables. 

Can you help with some ideas to decorate a rectangle living room?

Add a lot of color and a few differently shaped items to spice up your room.

If your room is a rectangle, focus on circular or rounded shapes. For example, a coffee table that has a curve like this would add some depth to a rectangle room.

When you have black and tan as your base, you are pretty free to add color where ever you see fit.  Put some bright red throw pillows on your couch. If red isn't your color, try a blue. Anything with a deep, vibrant tint will stand out.  Focus on different fabrics as well.

On your floor, add an area rug that has circular shapes such as this Funky Cream or this Dark Chocolate shag. A rug this color will contrast very well with your couch.

I'm not sure if you have windows in your room, but if you do you should add a curtain. Curtains always add a warmth to your room that bare windows are lacking.

If you can, grab an end table or two. Putting lamps on your end tables and surrounding your couch will make your room cozy. A vase with fresh flowers also adds a nice touch.