The resulting XML should contain only 2 attributes from "InspectionExt"
in case result="10" or "11" otherwise all attributes and "Obligations",
"Defects" but no "Photo"
Also when copying the Defects/Defect items, only those with
"not(@reinspection='ok'" should be copied over. No "Defects" element
should be created if nothing from source xml is copied.
The resulting "Defect" element must not contain the "reinspection"
attribute.

That's what I came up with after your suggestions:
<?xml version="1.0" encoding="UTF-8"?>
<xsl:stylesheet version="1.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
xmlns:xs="http://www.w3.org/2001/XMLSchema"
xmlns:fn="http://www.w3.org/2005/xpath-functions">
   <xsl:output method="xml" version="1.0" encoding="UTF-8" indent="yes"/>

   <xsl:template match="InspectionExt">
     <xsl:element name="ns2:InspectionReportInput"
namespace="http://somenamespace">
<xsl:choose>
<xsl:when test="@result eq '10' or @result eq '11'">

<xsl:copy-of select="@result"/>
<xsl:copy-of select="@expert"/>
</xsl:when>
<xsl:otherwise>
       <xsl:copy-of select="@*[not(@signed)]"/>
       <xsl:for-each select="Defects">
         <xsl:element name="Defects">
           <xsl:for-each select="Defect[not(@reinspection='ok')]">
             <xsl:element name="Defect">
               <xsl:copy-of select="@*[not(@reinspection)]"/>