Any criticism on my JavaScript micro-library? I made this little library for personal use to make javascripting faster and easier.
I would like to know if there are any inefficiencies in it (in size and performance) and possibly if there is a feature you think is missing that would be useful.

//PUT ALL CODE AT THE END OF THE PAGE!!!!!! THIS IS UBER IMPORTANT

//document shorthand
doc=document

//querySelectorAll shorthand
function $q(a){return doc.querySelectorAll(a)}

//returns specific element
function $(a,b){return $q(a)[b||0]}

//function to set properties in a chainable manner
Element.prototype.$p=function(a){for(b in a)this[b]=a[b];return this}

//creates a class that hides elements (notice the use of the function above)
$("head").appendChild(doc.createElement("style").$p({"innerHTML":".wqh{display:none}"}))

Element.prototype.$p({
"removeClass":function(a){this.classList.remove(a)},
"addClass":function(a){this.classList.add(a)},
"hide":function(){this.addClass("wqh")},
"show":function(){this.removeClass("wqh")},
//loads content of a file inside the element, example: element.load("random file.txt")
"load":function(a,b,c){c=this;b=new XMLHttpRequest;b.open("GET",a,!0);b.send();b.onreadystatechange=function(){c.innerHTML=b.responseText}}
})


Example of its uses:

$(".test").addClass("red")
$(".test",1).removeClass("big")
$(".test",2).hide()

for(a in b=$q("p"))
b[a].style.fontFamily="serif",
b[a].style.color="blue",
b[a].style.fontSize="20px";


More readable version of the $p() function:

//function to set properties in a chainable manner
Element.prototype.$p = function(proplist){
    var property;
    for(property in proplist){this[property] = proplist[property]}
    return this;
}
Problems with it : 


in the global namespace
use of innerHTML
no cross-compatibility 
messing with object prototypes
quite useless