You shoudn't have any downtime. Copy your current DNS records for your transferring domain(s) and input them at the destination Registrar, then initiate the transfer, and done. It is surprisingly easy.

The only edge case to check is if you are using non-standard or less common DNS record types (ALIAS, CAA) then first confirm with the destination Registrar that they support the record types you need.

Or if you want to keep things really simple, switch to a third-party DNS host before transferring and use the same name-server settings.

And yes, paying the target Registrar should extend your registration time. For example, say you bought a domain last year and paid to register it for five years (so four years remain). Now you transfer it to Gandi.net and pay their transfer fee. You used up one year originally, transferred, and it's yours for five years again. Your unused years at the outgoing Registrar 'roll over' to the new one.