Perl 5, 485 bytes

undef$/;$_=<>;s/\[code]((?:(?R)|.)*?)\[\/code]/push@a,$1;"\0"/geis;1while s{\[(b|i|u|img|url|color|size|quote)(?:=([^]]+))?](.*?)\[/\1]}{($n,$x,$y)=(lc$1,$2,$3);$n=~/^u/?$n=~/l/?"<a href=\"".($x||$y)."\">$y</a>":"<u>$y</u>":$n=~/^i/?$n=~/m/?"<img src=\"$y\">":"<em>$y</em>":$n=~/^b/?"<strong>$y</strong>":$n=~/^q/?"<blockquote>".($x?"<cite>$x</cite>":"")."$y</blockquote>":"<span style=\"".($n=~/c/?"color":"font-size").":$x\">$y</span>"}gie;s/\0/"<code>".shift(@a)."<\/code>"/ge;print

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Detailed explaination

undef$/;$_=<>;

Undefines the line separator so $_=<>; reads the entire input (newlines and all) into the default variable $_ at once.

s/\[code]((?:(?R)|.)*?)\[\/code]/push@a,$1;"\0"/geis;

Matches a [code] block and substitutes its contents with a null byte \0, saving them in the array @a for later.

1 while s{\[(b|i|u|img|url|color|size|quote)(?:=([^]]+))?](.*?)\[/\1]}{
  ($n,$x,$y)=(lc$1,$2,$3);
  $n=~/^u/?$n=~/l/?"<a href=\"".($x||$y)."\">$y</a>":"<u>$y</u>":
  $n=~/^i/?$n=~/m/?"<img src=\"$y\">":"<em>$y</em>":
  $n=~/^b/?"<strong>$y</strong>":
  $n=~/^q/?"<blockquote>".($x?"<cite>$x</cite>":"")."$y</blockquote>":
  "<span style=\"".($n=~/c/?"color":"font-size").":$x\">$y</span>"
}gie;

s{...}g replaces all matching tags in the string. The while loop handles nested tags.

• $n is the tag name
• $x are the tag attributes if present
• $y is what's within the tag

Inside the loop, a set of if/else replace the appropriate tags.

• Starts with u? -> u or url
• Starts with i? -> img or i
• Starts with b? -> b.
• Starts with q? quote (adds <cite>...</cite> only if an author was defined)
• Else: It must be color or size. Merge these as they both output <span style="..."> by checking if the name contains c to decide between color: or font-size:.

s/\0/"<code>".shift(@a)."<\/code>"/ge;
print

s/\0/.../ge finds every null byte placeholder created in the beginning and replaces it (FIFO) shift(@a) with the array contents.