With the Services widget I’d like to not have an icon, but an image of my choosing. How can I do this?
You can create your own custom field using Type plugin.
First, you have to go to Types > Custom fields > choose Service icon then choose image field from available fields http://prntscr.com/7i9coq, named it as service image for the field name and service-image for the field slug. Save and upload your image from Services.
Next, you need to open fp-services.php from appearance > editor then add this line (see this picture for the location of the new code http://prntscr.com/7i9e8y)
<?php $image = get_post_meta( get_the_ID(), 'wpcf-service-image', true ); ?> and
<?php elseif ($image) : ?> <img src="<?php echo esc_url($image); ?>" alt="<?php the_title(); ?>">