Improve header button spacing (#197896)

Closes #197873 

## Summary

- Fix spacing between 'Give feedback' and AI button.
- While in the neighborhood, I noticed the search button had some extra
padding, so I fixed that quick too.
- Lastly, I pushed the project switcher to the far right. This is only
used for local development of Serverless and was otherwise appearing
between production buttons.

### Before
<img width="420"
src="https://github.com/user-attachments/assets/f5abe1af-1762-4658-8040-d802b9752667"
/>

<img width="420"
src="https://github.com/user-attachments/assets/04288ff3-6012-4518-866f-0dea4ad62401"
/>

### After
<img width="420"
src="https://github.com/user-attachments/assets/ef2e0eff-d4f4-4ceb-bbf8-c39d2e3e9949"
/>

## QA
To test, you'll need to run this in serverless mode where the 'Give
feedback' button appears.
You can force it on by setting the following line equal to `true`.

ae9c0d3850/src/plugins/guided_onboarding/public/plugin.tsx (L50)

Co-authored-by: Elastic Machine <elasticmachine@users.noreply.github.com>

packages/core/chrome/core-chrome-browser-internal/src/ui/project/header.tsx

@@ -306,7 +306,11 @@ export const ProjectHeader = ({
                 />
               </EuiHeaderSectionItem>
 
-              <EuiHeaderSectionItem>
+              <EuiHeaderSectionItem
+                css={css`
+                  gap: ${euiTheme.size.s};
+                `}
+              >
                 <HeaderNavControls navControls$={observables.navControlsRight$} />
               </EuiHeaderSectionItem>
             </EuiHeaderSection>

x-pack/plugins/global_search_bar/public/components/search_bar.tsx

@@ -325,11 +325,12 @@ export const SearchBar: FC<SearchBarProps> = (opts) => {
         buttonRef={visibilityButtonRef}
         color="text"
         data-test-subj="nav-search-reveal"
-        iconType="search"
         onClick={() => {
           setIsVisible(true);
         }}
-      />
+      >
+        <EuiIcon type="search" size="m" />
+      </EuiHeaderSectionItemButton>
     );
   }
 

x-pack/plugins/serverless/public/plugin.tsx

@@ -61,7 +61,7 @@ export class ServerlessPlugin
       const { currentType } = developer.projectSwitcher;
 
       core.chrome.navControls.registerRight({
-        order: 500,
+        order: 5000,
         mount: (target) => this.mountProjectSwitcher(target, core, currentType),
       });
     }