[SalesForce] How to call a Content Block into an Email built with Content Builder

First, I'd move the retrieval inside your @card_type conditional and output each @contentArea value. Like this:

%%[
Var @card_type, @Row, @rows, @ac, @au, @ea, @se, @ContentArea1, @ContentArea2, @ContentArea3, @ContentArea4, @ContentArea5, @ContentArea6, @ContentArea7, @ContentArea8, @ContentArea9, @ContentArea10, @ContentArea11, @ContentArea12, @ContentArea13

SET @card_type = [CARD_TYPE]
SET @ac = [AC]
SET @au = [AU]
SET @ea = [EA]
SET @se = [SE]

IF Not Empty(@card_type) THEN

    SET @rows = LookupRows("forum-lookup-rows - 1","card_type", [CARD_TYPE])

    IF RowCount(@rows) > 0 THEN

        SET @row = Row(@rows,1)
        SET @ContentArea1 = Field(@row,"ContentArea1")
        SET @ContentArea2 = Field(@row,"ContentArea2")
        SET @ContentArea3 = Field(@row,"ContentArea3")
        SET @ContentArea4 = Field(@row,"ContentArea4")
        SET @ContentArea5 = Field(@row,"ContentArea5")
        SET @ContentArea6 = Field(@row,"ContentArea6")
        SET @ContentArea11 = Field(@row,"ContentArea11")

        IF @ac == 'n' THEN
            SET @ContentArea7 = Field(@row,"ContentArea7")
        ELSEIF @ea == 'n' THEN
            SET @ContentArea8 = Field(@row,"ContentArea8")
        ELSEIF @au == 'n' THEN
            SET @ContentArea9 = Field(@row,"ContentArea9")
            SET @ContentArea12 = Field(@row,"ContentArea12")
        /*ELSEIF @au == 'y' THEN
        SET @ContentArea13 = Field(@row,"ContentArea13")*/
        ELSEIF @se == 'n' THEN
            SET @ContentArea10 = Field(@row,"ContentArea10")
        ENDIF
    ENDIF

ENDIF

]%%

@card_type: %%=v(@card_type)=%%<br>
@ac: %%=v(@ac)=%%<br>
@au: %%=v(@au)=%%<br>
@ea: %%=v(@ea)=%%<br>
@se: %%=v(@se)=%%<br><br>
@ContentArea1: %%=v(@ContentArea1)=%%<br>
@ContentArea2: %%=v(@ContentArea2)=%%<br>
@ContentArea3: %%=v(@ContentArea3)=%%<br>
@ContentArea4: %%=v(@ContentArea4)=%%<br>
@ContentArea5: %%=v(@ContentArea5)=%%<br>
@ContentArea6: %%=v(@ContentArea6)=%%<br>
@ContentArea11: %%=v(@ContentArea11)=%%<br><br>
@ContentArea7: %%=v(@ContentArea7)=%%<br>
@ContentArea8: %%=v(@ContentArea8)=%%<br>
@ContentArea9: %%=v(@ContentArea9)=%%<br>
@ContentArea12: %%=v(@ContentArea12)=%%<br>
@ContentArea10: %%=v(@ContentArea10)=%%<br>

For any @ContentArea* field that's empty the email's going to blow up. You'll either need to have some default content for those or conditionally display alternate content in the body of the email. You might utilize an iif() function -- like this:

%%=iif(empty(@ContentArea1),"not found", ContentArea(@ContentArea1))=%%  

First, I'd suggest setting a primary key on your MatthewDataExtension Data Extension, since the ClaimRow() function is doing an update.

Also, every pair of ClaimRow() parameters after the 2nd needs to be the variable name and then the variable value -- in order to retrieve the proper row. So if you're going to update WebPromoCode and StoreCouponCode, your function would be this:

%%[

var @numberCustomer
var @WebPromoCode
var @StoreCouponCode
set @numberCustomer = AttributeValue("Customer_No")
set @WebPromoCode = AttributeValue("WebPromoCode")
set @StoreCouponCode = AttributeValue("StoreCouponCode")

set @couponDataExtensionRow = ClaimRow("MatthewDataExtension", "isClaimed", "WebPromoCode", @WebPromoCode, "StoreCouponCode", @StoreCouponCode , "Customer_No", @numberCustomer)

/* Snip */ 

]%%

If it helps, here's my go-to coupon claiming AMPScript:

%%[

var @couponRow, @couponCode, @rows, @row, @emailaddr

set @emailaddr = AttributeValue("emailaddr")

set @rows = LookupRows("couponCodes","EmailAddress",@emailaddr)

if rowcount(@rows) > 0 then
  set @row = Row(@rows,1)
  set @couponCode= field(@row,"couponCode")
else

    if _messagecontext == "PREVIEW" then
           set @couponCode = "XX TEST XX"
    else

      set @couponRow = ClaimRow("couponCodes","IsClaimed","EmailAddress",emailaddr) 

          if empty(@couponRow) then
             /* You can do other error handling here if you want.*/
             /* This aborts the send */
             raiseError("no more coupons available")
          else 
             set @couponCode= field(@couponRow,"couponCode") 
          endif

    endif 

endif

]%%

Here's your coupon code: %%=v(@couponCode)=%%

There are a few more details about the Data Extension on in my blog post on the subject.