Well, first off; read up on it first and then ask any remaining questions later - you'll be more likely to understand the answers then.
Second: a tag namespace attribute looks something like
<html xmlns:math="http://www.w3.org/1998/Math/MathML" xmlns="http://www.w3.org/TR/REC-html40">
which says that the default namespace is http://www.w3.org/TR/REC-html40 (which is the URI for HTML), but there are also tags from MathML (namespace URI http://www.w3.org/1998/Math/MathML). So this document (presumably) contains HTML tags written like <body>, <span>, etc., and MathML tags, written <math:mo>, <math:mrow>, etc.
The html tag could have been written with namespace attributes
<html xmlns:mathml="http://www.w3.org/1998/Math/MathML" xmlns:html="http://www.w3.org/TR/REC-html40">
and those tags would be <html:body>, <html:span>, <mathml:mo>, <mathml:mrow>, etc.
That said, one can switch default namespaces midstream, so that you don't have to specify the current namespace explicitly too often:
<table width="100%" class="equation"><tr><td>
<math xmlns="http://www.w3.org/1998/Math/MathML" class="equation">
<mspace id="x1-1002r2" class="label" width="0" /><mspace class="endlabel" />
<msup><mi>E</mi><mrow><mi>y</mi>
....
Note that the MathML section starts off with a <math> tag, which is the top-level tag for a MathML document, and comes with a namespace so's the client knows what it's and its contents are supposed to be ('cos <math> isn't HTML).
Thirdly, XML is case-sensitive for both tag and attribute names. It's xmlns, not XMLNS.
Fourthly, regarding the "Also, does the "HTML" part, do away with the original opening <html> tag? ": The "'HTML' part" is the <html> tag, with attributes.
Fifthly, HTML is not an XML dialect. If you want that, you're after XHTML. There are differences (which is how XHTML is defined).