I want to add the alt attribute to all the photos!

If your current code manages all blocks with images replacing usage wp_get_attachment_image_src with output of hardcoded img tag, with echo wp_get_attachment_image() which will return img tag as string with default alt of image or empty if non set will work.

Below is your code updated with idea described.

<?php
class home_main_post_below_list extends AQ_Block {
    //set and create block
    function __construct() {
        $block_options = array(
            'name' => 'Home main post with below list post',
            'size' => 'span12',
        );

        //create the block
        parent::__construct('home_main_post_below_list', $block_options);
    }
    //create form
    function form($instance) {
        $titles = isset($instance['titles']) ? esc_attr($instance['titles']) : 'Recent Posts';
        $number_show = isset($instance['number_show']) ? absint($instance['number_show']) : 5;
        ?>
        <p><label for="<?php echo $this->get_field_id('titles'); ?>"><?php _e('Title:', 'tl_back'); ?></label>
            <input class="widefat" id="<?php echo $this->get_field_id('titles'); ?>" name="<?php echo $this->get_field_name('titles'); ?>" type="text" value="<?php echo $titles; ?>" /></p>
        <p><label for="<?php echo $this->get_field_id('number_show'); ?>"><?php _e('Number of posts to show:', 'tl_back'); ?></label>
            <input id="<?php echo $this->get_field_id('number_show'); ?>" name="<?php echo $this->get_field_name('number_show'); ?>" type="text" value="<?php echo $number_show; ?>" size="3" /></p>

        <label for="<?php echo $this->get_field_id('cats'); ?>"><?php _e('Select categories to include in the recent posts list:', 'tl_back'); ?>

            <?php
            $categories = get_categories('hide_empty=0');
            echo "<br/>";
            foreach ($categories as $cat) {
                $option = '<input type="checkbox" id="' . $this->get_field_id('cats') . '[]" name="' . $this->get_field_name('cats') . '[]"';
                if (isset($instance['cats'])) {
                    foreach ($instance['cats'] as $cats) {
                        if ($cats == $cat->term_id) {
                            $option = $option . ' checked="checked"';
                        }
                    }
                }
                $option .= ' value="' . $cat->term_id . '" />';
                $option .= '&nbsp;';
                $option .= $cat->cat_name;
                $option .= '<br />';
                echo $option;
            }
            ?>
        </label>
        <?php

    }

    //create block
    function block($instance) {

        extract($instance);

        $titles = apply_filters('widget_title', empty($instance['titles']) ? 'Recent Posts' : $instance['titles'], $instance, $this->id_base);


        if (!isset($instance["cats"])) {
            $cats="";}

        // array to call recent posts.

        $themeloy_args = array(
            'showposts' => $number_show,
            'category__in' => $cats,
        );

        $themeloy_widget = null;
        $themeloy_widget = new WP_Query($themeloy_args);

        ?>
        <div class="widget post_list_medium_widget">
            <?php if (!empty($instance['titles'])) {?><h3 class="widget-title"><span><?php echo $instance["titles"];?></span></h3><?php }?>
            <div class="widget_container">
                <div class="post_list_medium">
                    <?php
                    $i = 0;
                    while ($themeloy_widget->have_posts()) {
                        $i++;
                        $themeloy_widget->the_post();
                        $post_id = get_the_ID();
                        $thumb = themeloy_get_thumbnail(get_the_ID());
                        if ($i == 1) {
                            ?>

                            <div class="list_item">

                                <div class="entry-thumb feature-item">
                                    <a href="https://wordpress.stackexchange.com/questions/360692/<?php the_permalink(); ?>">
                                        <?php if ( has_post_thumbnail()) {
                                            echo wp_get_attachment_image( get_post_thumbnail_id( $post_id ), 'feature-large', false, array('class'=>'cat-feature-large-img') );
                                        } else {echo '<img class="cat-feature-large-img" src="'.get_template_directory_uri().'/images/demo/feature-large.jpg'.'" alt="">';} ?>
                                    </a>

                                    <div class="cat-feature-large">
                                        <?php $post_cat = get_post_custom_values('cat_themeloy_select', get_the_ID());
                                        $post_cat = ($post_cat[0] != '') ? $post_cat[0] : of_get_option('blog_category_post');
                                        if($post_cat == '1'){echo '<p class="cat-slider">'; echo the_category(', ').'</p>';}?>
                                    </div>

                                    <?php echo themeloy_post_type(); ?>
                                </div>
                                <h2><a href="https://wordpress.stackexchange.com/questions/360692/<?php the_permalink(); ?>" class="title"><?php the_title(); ?></a></h2>

                                <?php echo themeloy_post_meta(get_the_ID());?>
                                <p><?php echo themeloy_short_title(320, get_the_excerpt('')); ?>  </p>
                            </div>
                            <div class="clear margin-buttons"></div>
                        <?php }else{?>

                            <div class="small-list-content">
                                <?php
                                $post_date = get_post_custom_values('date_themeloy_select', get_the_ID());
                                $post_date = ($post_date[0] != '') ? $post_date[0] : of_get_option('blog_date_post');
                                if($post_date == '1'){
                                    ?>
                                    <div class="feature_post_style">
                        <span class="post_date"><span class="date_number"><?php echo get_the_date('d');?></span> <?php echo get_the_date('M');?>
                        <i class="icon-caret-right feature-icon-right"></i>
                        </span>
                                        <span class="post_time"><i class="icon-time"></i> <?php echo get_the_time('H:i');?></span>
                                    </div>
                                <?php }?>
                                <div class="feature-link feature-item">
                                    <a href="https://wordpress.stackexchange.com/questions/360692/<?php the_permalink(); ?>" class="feature-link">
                                        <?php if ( has_post_thumbnail()) {
                                            echo wp_get_attachment_image( get_post_thumbnail_id( $post_id ), 'medium-feature', false );
                                        }else{echo '<img src="'.get_template_directory_uri().'/images/demo/medium-feature.jpg" alt="">';} ?>
                                    </a>
                                    <?php echo themeloy_post_type(); ?>
                                </div>

                                <div class="list_desc">
                                    <h4 class="list_title"><a class="post-title" href="https://wordpress.stackexchange.com/questions/360692/<?php the_permalink(); ?>" rel="bookmark" title="<?php the_title_attribute(); ?>">
                                            <?php the_title(); ?>
                                        </a></h4>

                                    <?php echo themeloy_post_meta(get_the_ID());?>

                                </div>
                                <div class="clear"></div>
                            </div>

                        <?php }}?>


                </div>
            </div>
        </div>

        <?php
        wp_reset_query();

    }

    function update($new_instance, $old_instance) {
        return $new_instance;
    }


}

Also fallback img tags if none post_thumbnail set got alt attribute empty as it’s required by accessibility guidelines.