External Share for Confluence

Can we display same results in search as Atlassian?

No. There is no API endpoint to do that.
We use CQL search to emulate experience but result can be different and have different order.
We suggested to Atlassian improvement: https://jira.atlassian.com/browse/CONFCLOUD-75118
Please vote to help us improve search experience.

Can we display custom emoji panel?
No. This information is not available via API endpoint.
We suggested to Atlassian improvement: https://jira.atlassian.com/browse/CONFCLOUD-73225