Magento – Magento 2 – How to remove the white borders from images in product list page

Set transparency to false.
You can read more about how to configure the image properties in view.xml at http://devdocs.magento.com/guides/v2.0/frontend-dev-guide/themes/theme-images.html

<transparency>false</transparency>

You have to override core file into your theme, Category layered navigation comes form core file,

vendor/magento/module-layered-navigation/view/frontend/templates/layer/view.phtml

You have to get files from core and keep in your theme,

app/design/frontend/{Vendor}}/{themename}}/Magento_LayeredNavigation/templates/layer/view.phtml

Now you can set condition for this,

<?php  if($filter->getName() != __('Category')) { } ?>

in code.

<?php

/**
 * Category layered navigation
 *
 * @var $block \Magento\LayeredNavigation\Block\Navigation
 */
?>
<?php if ($block->canShowBlock()): ?>
    <div class="block filter">
        <div class="block-title filter-title">
            <strong><?php /* @escapeNotVerified */ echo __('Shop By') ?></strong>
        </div>

        <div class="block-content filter-content">
            <?php echo $block->getChildHtml('state') ?>

            <?php if ($block->getLayer()->getState()->getFilters()): ?>
                <div class="block-actions filter-actions">
                    <a href="<?php /* @escapeNotVerified */ echo $block->getClearUrl() ?>" class="action clear filter-clear"><span><?php /* @escapeNotVerified */ echo __('Clear All') ?></span></a>
                </div>
            <?php endif; ?>
            <?php $wrapOptions = false; ?>
            <?php foreach ($block->getFilters() as $filter): ?>
                <?php if (!$wrapOptions): ?>
                    <strong role="heading" aria-level="2" class="block-subtitle filter-subtitle"><?php /* @escapeNotVerified */ echo __('Shopping Options') ?></strong>
                    <dl class="filter-options" id="narrow-by-list">
                <?php $wrapOptions = true; endif; ?>
                <?php  if($filter->getName() != __('Category')) { ?>
                    <?php if ($filter->getItemsCount()): ?>
                        <dt role="heading" aria-level="3" class="filter-options-title"><?php echo $block->escapeHtml(__($filter->getName())) ?></dt>
                        <dd class="filter-options-content"><?php /* @escapeNotVerified */ echo $block->getChildBlock('renderer')->render($filter); ?></dd>
                    <?php endif; ?>
                    <?php } ?>
            <?php endforeach; ?>
            <?php if ($wrapOptions): ?>
                </dl>
            <?php endif; ?>
        </div>
    </div>
<?php endif; ?>