heapify last 3 lines I'm following 
http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-006-introduction-to-algorithms-fall-2011/lecture-videos/MIT6_006F11_lec04.pdf

Last 3 lines of heapify pseudocode are:

if largest =! i
    then exchange A[i] and A[largest]
    Max_Heapify(A, largest)


What is the aim of the last line?
Since largest still holds value of largest leaf, which is now root, what would the recursive run achieve? Just an empty run, it seems.Max-heap is already built.
You are asking several related questions. I will only answer the first one. The heapify procedure gets an almost-heap $A$ and a node $i$. The heap property is satisfied for all nodes except for $i$. Heapify proceeds as follows:


Determine the largest child $j$ of $i$.
If $A[i] \geq A[j]$, the heap property is already satisfied.
Otherwise, exchange $A[i]$ and $A[j]$, and run heapify on $A$ and $j$.


After the exchange, the heap property is no longer violated for $i$, but might be violated for $j$. For this reason we have to call heapify recursively on the node $j$.

I suggest you try a few examples and see for yourself that sometimes the recursive call is necessary.