I’m working on a free Chrome New Tab extension that helps users learn Korean and integrates a basic search feature. The extension will work similarly to the normal Google search experience:

Users type a query in the search bar on the new tab page.
Autocomplete suggestions appear as they type to enhance usability.
When users submit the query, they are redirected to the standard Google search results page (e.g., https://www.google.com/search?q=query).
To provide autocomplete suggestions, I’m currently fetching data from Google’s endpoint:
https://www.google.com/complete/search?client=chrome&q=query
I’m using a proxy server to fetch and return the suggestions to the extension.

Before proceeding, I’d like to clarify the following:

Is fetching autocomplete suggestions this way compliant with Google’s terms of service for a Chrome extension?
If not, what is the recommended approach for providing autocomplete suggestions while keeping the experience similar to Google’s?
Would Google Programmable Search Engine (PSE) be applicable here, even though we’re redirecting users to Google’s unmodified search results page?
I want to ensure this implementation follows best practices and remains fully compliant. Any guidance or recommendations would be greatly appreciated.