Is there a way to send unconfirmed transactions and get them accepted by the network? Something like in bitcoin, where if you send a transaction that is using unconfirmed inputs, the outputs will only be confirmed when the inputs are?
No. Monero does not support chained payments within the same block. Outputs must be available before they can be spent. Because of how ring signatures work, Monero is not easily comparable to Bitcoin in this regard.
If not, could you tell me what's the best way to speed up the time it takes to have the newly received funds ready to be sent?
You cannot speed up confirmation time unless you have access to a huge amount of hashing power which you can activate on demand to (temporarily) speed up block discovery until the network difficulty adjusts.
A better solution would be to have a Monero wallet that has a balance that is always available to you. In that case you can spend that amount of Monero at any time without needing to wait for incoming transactions to be confirmed first. If the incoming transaction is from a third party keep in mind the security implications of accepting transactions that have not yet been accepted by the network.
Would using the lowest possible mixin value have an influence?
Lastly, what is Monero's confirmation time? It is in blocks, right?
Yes. Each block is approximately 2 minutes apart (on average) but incoming transactions require 10 blocks before the funds can be spent, regardless of how long those ten blocks take.