The open and close [raw] shortcode is showing because the theme you’re using does not include it and by viewing your screenshot, I don’t think you need it.
However, the theme is overriding the styles of the search button. The theme also seems to be overriding the width of the category select.
The character index is working as designed, it does not filter by character, it will only jump to the start of the character group if it is on the current results page.
When you perform a search, look in the address bar. You should see
cn-s, do you see the word “Search” in addition to the name you’re searching for? If so, then the theme, most likely, is causing the issue.
Unfortunately, with just a screenshot, I can provide any solutions, I’ll need a link to you directory.