How To Read HTML code as XML and get the output like the sample in sql?

I am trying to establish communication between nodesh4 and ul.

You can use the << and >> operator to check if a node is before or after another node in document order. Combine that with a predicate on position, [1], to get the first occurrence also in document order.

select H4.X.value('(span/text())[1]', 'varchar(10)') as Section,
       UL.X.query('.') as UL
from @X.nodes('/div/section/h4') as H4(X)
  cross apply H4.X.nodes('(let $h4 := . (: Save current h4 node :)
                           return /div/section/ul[$h4 << .])[1]') as UL(X);

rextester:

<< and >> are called Node Order Comparison Operators

If you have an XML fragment like this:

<N1>1</N1>
<N2>2</N2>
<N3>3</N3>
<N4>4</N4>
<N5>5</N5>

you can get all nodes before the first occurrence of N3 with this query:

select @X.query('/*[. << /N3[1]]');

Result:

<N1>1</N1>
<N2>2</N2>

/* will give you all root nodes. What is enclosed in [] is a predicate. . is the current node and /N3[1] is the first N3 node in document order at the root level. So from each root node you get the nodes that precede N3.

Here is almost the same query, only you get the nodes that follow the first N3 node:

select @X.query('/*[. >> /N3[1]]');

<N4>4</N4>
<N5>5</N5>

To only get the first node after the first N3 node, you add the predicate [1]:

select @X.query('/*[. >> /N3[1]][1]');

<N4>4</N4>

This is not exactly elegant but seems to do the job.

DECLARE @X XML = REPLACE(REPLACE(@S, '<h4>', '<foo><h4>'), '</ul>', '</ul></foo>')

SELECT Category = x.value('../../h4[1]/span[1]', 'varchar(10)'),
       Selection = x.value('descendant-or-self::text()[1]', 'varchar(10)'),
       Value = REPLACE(
                REPLACE(
                 REPLACE(
                  LTRIM(
                   RTRIM(
                    REPLACE(
                     REPLACE(
                      CAST(x.x.query('fn:data(descendant-or-self::text()[fn:position() > 1])') AS VARCHAR(MAX))
                     , char(10), '')
                    , char(13), '')
                   )
                  )
                 , '  ', ' |')
                , '| ', '')
               , '|', '')
FROM   @X.nodes('div/section/foo/ul/li') x(x)
ORDER  BY Category,
          Selection

Which returns

+----------+-----------+-------+
| Category | Selection | Value |
+----------+-----------+-------+
|  A       |  Ab       | AD AC |
|  A       |  Ag       | AL    |
|  B       |  Bb       | BD BC |
|  B       |  Bg       | BL    |
+----------+-----------+-------+

I'm assuming this is what you want as the desired results table in the question does not return the "rest of the values as a concatenated string"