Thanks for reaching out.
First, please check Ethos documentation here https://theme.co/apex/forum/t/customizer-ethos-options/131
1) Category IDs are automatically generated in database, so you have to update the IDs yourself with the existing one. Please check the Filterable Index from the above URL.
And please check this https://community.theme.co/kb/how-to-locate-category-ids/ to help you finding your preferred category IDs.
2) Please check the section Post Carousel from the above URL. There are three options, most commented, random, and featured. If you select the
featured one then you must manually select a post to be featured. And you can do that by editing the post in admin and find this section
Check the option where you wish to display that current post. The order is based on comments, or random, or by date if it's featured.
3) It can't be set as full-width because of its container, what you could do is just disable the post slider and use revolution slider. Please check this https://theme.co/apex/forum/t/extension-revolution-slider/71. And this another thread so you could implement it on blog page.
4) They are widgets, please go to Theme Options > Layout and Design > CONTENT LAYOUT and make sure it's set to
Content Sidebar instead of full-width. Then go to Theme Options > Blog and make sure all layout are set to Global instead of Full-width, to make sure it inherit the layout from Layout and Design. Then go to Admin > Appearance > Widgets and add Search widget, Recent posts widget, and Calendar widget to the Main Sidebar.
5) Are you referring to video post format? Yes, you can add Youtube video but as an Embed code instead of just youtube URL. Please check this https://support.google.com/youtube/answer/171780?hl=en