Regarding question #1, no, you did not make an excess contribution. You recharacterized $6,000, not $8,400.
Regarding the $2,400 you withdrew, did you request a return of contribution (resulting in a gain/loss-adjusted amount being distributed or did you request a regular distribution?
If you requested a return of contribution of $2,400, it seems like somewhere around $3,360 would have been distributed to you, including $960 of gains that would be taxable. In that case you have contributed a net amount for 2022 of $6,000 - $2,400 = $3,600 and you have until April 18, 2023 to contribute a new $2,400 to make your contribution for 2022 be a full $6,000.
If you requested a regular distribution of $2,400 and you received $2,400, you have a distribution that is eligible for rollover provided that it would not result in a violation of the one rollover-per-12-months limitation.
