No, Gandimail does not support DKIM.
If you use Gandi's DNS however, you are free to add the TXT zone file record needed to add the necessary “_domainkey” TXT entry for your domain. You would then need to see with your outgoing email provider concerning the configuration of their mail server for the service.
Technically, yes, but you may encounter problems with some email not being received when sending “from” a forwarding address. Since it is not technically possible for email to originate from a forwarding address, such messages are considered by some mail servers to be “spoofed” and therefore block them.
You may therefore find it easier to replace the forwarding addresses with an actual mailbox (with aliases, if necessary), then configure your email client to be able to send and receive from that address.
You can use Gandi Mail with your domain regardless of what type of hosting you have. You can even use it without having any hosting at all, though you may want to upgrade to one of the Mail Packs depending on your usage.
No. Please see Use and configure GandiMail for instructions.
No. GandiMail can not be used with a domain not registered with Gandi.