How can I set omnichannel routing to auto-open tickets?

Can I set omnichannel routing to automatically set ticket statuses to Open?

Yes, you can set omnichannel routing to automatically accept and open tickets for email and messaging in the omnichannel routing configuration page:

For messaging tickets, select Auto-accept for an agent

For email tickets, select Automatically open assigned email tickets

Note: Voice or Talk tickets don't support omnichannel routing. Agents must accept these tickets manually.

See this article for more information: Turn on automatic conversation acceptance for agents.