2. Featured posts images now are just smaller versions of full resolution files, which sometimes look very bad (image for post “Okna energooszczÄ™dne Schuco CT70” in 4-column-grid). How to force certain resolution for featured posts images (for example: 500 x 300px)?
That is necessary in order to cater to the wide variety of possible image sizes, and the different screen sizes of devices the site is displayed on. For example, you might have 4 columns displayed on desktop, but on a phone there will only be one column. While the image size for the 4-column desktop can be quite low in resolution, this would result in blurry images in modern phones with high density screens.
The image sizes that the theme uses are a compromise of these factors. You may be able to use a plugin to override the theme’s featured image plugin, but be aware that your images might not look the best on every device.