Can we place `<img>` inside `<h1>` according to web standards?

Yes, you can - the DTD says:

<!ELEMENT h1  %Inline;>
<!ENTITY % Inline "(#PCDATA | %inline; | %misc.inline;)*">
<!ENTITY % inline "a | %special; | %fontstyle; | %phrase; | %inline.forms;">
<!ENTITY % special "%special.pre; | object | img ">

That basically means h1 can contain %Inline, which is made of various things, including img

Look who is using it: http://www.w3.org/

<h1 class="logo"><a tabindex="2" accesskey="1" href="/"><img src="/2008/site/images/logo-w3c-mobile-lg" width="90" height="53" alt="W3C" /></a> <span class="alt-logo">W3C</span></h1>