Randomize the test

Pyth - 48 42 41 39 bytes

Will packed string.

js.Sm++hd/D"None of the above".Stdkc.zk

Try it online here.

JavaScript ES6, 170 bytes

Is an anonymous function, name it. Note: this uses the random sort method, which isn't entirely random, but is sufficient, provided you aren't a probability teacher.

t=>t.split`

`.map(x=>(x=x.split`
`,R=[],(k=x.pop())=="  None of the above"?(R=[k]):x.push(k),[x.shift(),...x.sort(r=_=>.5-Math.random()),...R].join`
`)).sort(r).join`

`

With the bonus, 180.75 bytes

t=>t.split`

`.map(x=>(x=x.split`
`,R=[],(k=x.pop())=="  None of the above"?(R=[k]):x.push(k),[x.shift(),...x.sort(r=_=>.5-Math.random()),...R].map((k,i)=>(i?`  ${" ABCD"[i]}. `:"")+k.trim()).join`
`),a=0).sort(r).map(e=>++a+". "+e).join`

`

Test it out!

F=t=>t.split`

`.map(x=>(x=x.split`
`,R=[],(k=x.pop())=="  None of the above"?(R=[k]):x.push(k),[x.shift(),...x.sort(r=_=>.5-Math.random()),...R].join`
`)).sort(r).join`

`;

g.onclick=function(){o.innerHTML = ""; o.appendChild(document.createTextNode(F(i.value)));}
g.click();

textarea{width:100%;height:14em;}textarea,div,button{font-family:Consolas,monospace;white-space:pre;}

<textarea id=i>When was the War of 1812?
  1812.5
  1776
  1812
  1821

What is the capital of France?
  46
  F
  Nice
  None of the above

What is the square root of -1 (simplified)?
  i
  i don't know
  square root of -1
  wait this is a math test?</textarea><button id=g>go &rarr;</button><div id=o></div>

CJam, 54 53 55 52 bytes

Saved 1 byte from using a later release (avaliable on TIO). Gained 2 bytes because I forgot to randomize order of questions. Saved 2 bytes from yet another bug fixed on TIO.

qNN+/mr{N/(\mr_"  None of the above"#3e\N*N\++}%NN+*

Try it online!