With some help from Firebug, I have been fortunate to find a solution on the web that I was able to implement.
If anyone else comes across this thread with the same issue at hand, here is how I resoved it:
<footer id="colophon" role="contentinfo">
<table width="100%" cellspacing="0" cellpadding="0" border="0">
<a href="http://wordpress.org/" title="Semantic Personal Publishing Platform">Proudly powered by WordPress</a><!-- .site-info -->
<p style="text-align: right;">
<a title="Profile for cresun on last.fm" href="https://www.last.fm/music/cresun" target=_"blank"><img src="http://cresun.is-great.net/wp-content/uploads/lastfm_badge_16x16.png" class="footer-badges"></a>
<a title="Listen to my music on SoundCloud" href="https://soundcloud.com/cresun" target=_"blank"><img src="https://w.soundcloud.com/icon/assets/images/orange_white_16-94fc761.png" class="footer-badges"></a>
To elaborate, I created a new footer.php file in my child theme directory, containing (only) the code listed above. If I understand this correctly, the footer.php of the child theme overrides the parent footer.php.
The HTML code was disclosed by analysing the footer on my website with Firebug, so it was a simple copy / paste operation to create the new footer.php. I then created a table with one row and two columns, so that the WordPress credit note would display in one cell, and the badges in the other. Also note that the second cell has a paragraph style element to align the badges to the right.
As for the visual styling, I learned how to define a class attribute for the specific images used, which is expressed at the end of the image elements. This class can then be styled through the styles.css file of the child theme. I did this the following way:
/* Styling for the footer badges */
opacity: 0.5; /* Sets default opacity value */
opacity: 1; /* Makes the images fully opaque */
In this example I copied some CSS code that I found on Stack Overflow, and replaced the class from the example with the one I had defined in my footer.php file. It makes my badges appear translucent by default, and fully opaque on hover. I also added a vertical-align property with a middle declaration, to make the badges align more evenly with the text on the left of the footer.
Now that the basics have been set, I'll be looking into how to make the badges gray by default and load the coloured images on hover, to better match my theme.