You get an email without a DKIM signature from me.

Is that because I do not use DKIM and do not sign my emails or because the message is forged and not from me at all?

You cannot tell. There is a single SPF record that you can look up to see if SPF is configured. It's just a TXT record on the domain. But there is no standard DKIM record. A DKIM signed email tells you what key identifier it used and you have to use that to look up the DKIM key at <key identifier>._domainkey.<sender domain>. When the email has no DKIM signature you cannot tell if that's because the sender domain does not use DKIM or if it does have a DKIM key record but it was not used, potentially because the email is forged.