Is long division the most optimal "effective procedure" for doing division problems? (Motivation: I am going to be working with a high school student next week on long division, which is a subject I strongly dislike.)
Consider:  $\frac{1110}{56}=19\frac{46}{56}$.
This is really a super easy problem, since once you realize $56*20=1120$ its trivial to write out $1110=56*19+46$.
You can work out the long division for yourself if you want; needless to say it makes an otherwise trivial problem into a tedious, multi-step process.
Long division is an "effective procedure", in the sense that a Turing machine could do any division problem once it's given the instructions for the long division procedure.  To put it another way, an effective procedure is one for which given any problem of a specific type, I can apply this procedure systematically to this type of problem, and always arrive at a correct solution.
Here are my questions:
1) Are there other distinct effective procedures for doing division problems besides long division?
2) Is there a way to measure how efficient a given effective procedure is for doing division problems?
3) Does there exist an optimal effective procedure for division problems, in the sense that this procedure is the most efficient?

The OP asks

Are there other distinct effective procedures for doing division
  problems besides long division?

You can certainly tweak the long division algorithm to make it better. Here, we will describe one such 'tweak improvement', motivated by the OP's sample problem.
Problem: Divide $1,110$ by $56$.
$560 \times 1 = 560 \le 1,110 \; \checkmark \; \text{OVERSHOOT: } 560 \times 2 = 1,120 \text{, } 1,120 - 1,110 = 10$
Algorithm detects that the overshoot is less than or equal to $56$, so takes this  path:
$\frac {1,110}{560} = 2 - \frac{10}{560} \text{ iff }$
$\tag ! \frac {1,110}{56} = 20 - \frac{10}{56} = 19 + \frac{(56-10)}{56} = 19 + \frac{46}{56}$
To better understand this approach, please see this link; it describes a way of organizing your $\text{Base-}10$ long division work that is amenable to  tweaks.
For the purpose of comparison, we solve the problem without the above 'one off' enhancement:
$560 \times 1 = 560 \le 1,110 \; \checkmark \; \text{OVERSHOOT: } 560 \times 2 = 1,120 \gt 1,110$
$\frac {1,110}{560} = 1 + \frac{550}{560} \text{ iff }$
$\tag 1 \frac {1,110}{56} = 10 + \frac{550}{56}$
$56 \times 9 = 504 \le 550 \; \checkmark  \; \text{OVERSHOOT: } 560$
$\tag 2 \frac {550}{56} = 9 + \frac{46}{56}$ 
Combining (1) and (2), 
$\tag 2 \frac {1,110}{56} = 19 + \frac{46}{56}$
Here we would have to build the entire multiplication table $56 \times n \; | \; 1 \le n \le 9$. 

I'm fairly confident that an algorithm could be created that would be able to divide by $D$ with only a $D \times n \; | \; 1 \le n \le 5$ table. The table could be expanded as necessary, using addition to get us up to, if necessary, $D \times 5$.
