Correct word to use for a suburban fence?
On the Hudong Baike page of 篱笆, 


  又叫栅栏、护栏，用来保护院子的一种设施，一般都是由棍子、竹子、芦苇、灌木或者石头构成，在我国北方农村很常见。


That means, 篱笆 is a kind of facility to protect private yards, which is usually made of wood sticks, bambooes, reeds, shrubs or stones. It is common in rural areas in Northern China.

So 篱笆 should be the most correct word to describe a fence.

And 栏杆, 


  栏杆是建筑物的楼、台、廊、梯等边沿处的围护构件，具有防护功能，兼起装饰作用。


So it's related to buildings (although it can be in other places, like a path in a mountain). 

Google Translate translates 栏杆 into railing, not fence.

Update:

The measure word, well, maybe 一排篱笆 will work. But as a Chinese I can't remember any measure word that is special for 篱笆...