Leaderboard golf

Python 3, 860 856 bytes

Golfed slightly, just to bootstrap the leaderboard and provide some template for other golfers:

import json,re,html as h,requests as r
p=print
u=h.unescape;a=[];n={}
for i in json.loads(r.get("https://api.stackexchange.com/2.2/questions/111735/answers?page=1&pagesize=100&site=codegolf&filter=!t)IWYnsLAZle2tQ3KqrVveCRJfxcRLe").text)["items"]:
    m=re.match(r'<h\d>\s*([^\n,]*[^\s,]),.*?(\d+)(?=[^\n\d<>]*(?:<(?:s>[^\n<>]*<\/s>|[^\n<>]+>)[^\n\d<>]*)*<\/h\d>)',i["body"].splitlines()[0]);l=u(m.group(1));t=u(i["owner"]["display_name"]);s=m.group(2);a.append((t,l,s))
    if l not in n: n[l]=[]
    n[l].append((t,s))
p("Leaderboard\n\tAuthor\tLanguage\tSize")
z=0;y=None
for i in enumerate(sorted(a,key=lambda x:x[2])):
    if y==i[1][2]:z+=1
    else:z=0;y=i[1][2]
    p("%d.\t%s\t%s\t%s"%(i[0]+1-z,i[1][0],i[1][1],i[1][2]))
p("Winners by Language\nLanguage\tUser\tScore")
for i in n.keys():
    n[i].sort(key=lambda x:x[1])
    print("%s\t%s\t%s"%(i,n[i][0][0],n[i][0][1]))

Indented with tabs. The last print is deliberately not replaced by p to create a score tie with the Mathematica answer.

Ungolfed:

import json
import re
import html
import requests
url="https://api.stackexchange.com/2.2/questions/111735/answers?page=1&pagesize=100&site=codegolf&filter=!t)IWYnsLAZle2tQ3KqrVveCRJfxcRLe"
data=json.loads(requests.get(url).text)
answers=[]
languages={}
for i in data["items"]:
    header=i["body"].splitlines()[0]
    m=re.match(r'<h\d>\s*([^\n,]*[^\s,]),.*?(\d+)(?=[^\n\d<>]*(?:<(?:s>[^\n<>]*<\/s>|[^\n<>]+>)[^\n\d<>]*)*<\/h\d>)', header)
    lang=html.unescape(m.group(1))
    author=html.unescape(i["owner"]["display_name"])
    score=m.group(2)
    answers.append((author, lang, score))
    if lang not in languages: languages[lang]=[]
    languages[lang].append((author, score))
answers.sort(key=lambda x:x[2])
print("Leaderboard")
print("\tAuthor\tLanguage\tSize")
rankadj=0
prevscore=None
for i in enumerate(answers):
    if prevscore == i[1][2]:
        rankadj+=1
    else:
        rankadj=0
        prevscore=i[1][2]
    print("%d.\t%s\t%s\t%s" % (i[0]+1-rankadj, i[1][0], i[1][1], i[1][2]))
print("Winners by Language")
print("Language\tUser\tScore")
for i in languages.keys():
    w=languages[i]
    w.sort(key=lambda x:x[1])
    print("%s\t%s\t%s" % (i, w[0][0], w[0][1]))

Note: it does not yet handle links correctly, so fails for, for example, question 17005.

Perl + Mojolicious, 468 456 469 504 bytes

Using Mojolicious library.

use v5.10;use ojo;while(@i=@{(g("http://api.stackexchange.com/2.2/questions/111735/answers?site=codegolf&filter=withbody&page=".++$p)->json//{})->{items}}){push@r,[$_->{owner}{display_name},(($h=x($_->{body})->at("h1,h2")||next)->at("a")||$h)->text=~/\s*([^,]+)\s*/,$h->text=~/(\d+)[^\d]*$/]for@i}$,="   ";say"Leaderboard
",Author,$l=Language,Size;say+(++$i,$s{@$_[2]}//=$i).".",@$_
for@r=sort{@$a[2]-@$b[2]}@r;%h=map{@$_[1],$_}reverse@r;say"Winners by $l
$l",User,Score;say$_,$h{$_}[0],$h{$_}[2]for keys%h

Ungolfed:

use v5.10;
use ojo;

my @r;
while (my @i = @{ (g("http://api.stackexchange.com/2.2/questions/111735/answers?site=codegolf&filter=withbody&page=" . ++$p)->json // {})->{items} }) {
    my $h = x($_->{body})->at("h1,h2") or next;
    push(@r, [$_->{owner}{display_name}, ($h->at("a") || $h)->text =~ /\s*([^,]+)\s*/, $h->text =~ /(\d+)[^\d]*$/]) for @i;
}

$, = "\t";
my %s;
say("Leaderboard\n", "Author", (my $l = "Language"), "Size");
say((++$i, $s{$_->[2]} //= $i) . ".", @$_) for @r = sort { $a->[2] <=> $b->[2] } @r;

my %h = map { $_->[1] => $_ } reverse(@r);
say("Winners by $l\n$l", "User", "Score");
say($_, $h{$_}[0], $h{$_}[2]) for keys(%h);