Does this mean my application has been rejected, has processed without the requested document?
It certainly doesn't mean that your application has been rejected. It does seem as though they are proceeding with the processing of your application without waiting for the requested document, and that does not bode well for the eventual outcome of the application.
However there's not much you can do. You can notify them that you still intend to supply the requested document but that you don't yet have it. Aside from that, you can wait until you receive either the response from your application or the document. If you receive the document first, forward it to the visa application center, of course.
Since you say you applied two months ago, I suspect that they are acting pursuant to Article 23(3) of the Schengen Visa Code, which provides that the deadline for processing a visa application may be extended to a maximum of 60 days if additional documents are requested. If this is the case, they have decided that since you haven't been able to provide the document within the deadline, they are going to process the application without the document.
If they refuse to issue the visa, you will probably have to submit a new application after you have received the document.
For reference, here are the first three of Article 23's four sections:
Decision on the application
Applications shall be decided on within 15 calendar days of the date of the lodging of an application which is admissible in accordance with Article 19.
That period may be extended up to a maximum of 30 calendar days in individual cases, notably when further scrutiny of the application is needed or in cases of representation where the authorities of the represented Member State are consulted.
Exceptionally, when additional documentation is needed in specific cases, the period may be extended up to a maximum of 60 calendar days.
The fourth point describes the possible outcomes of the application. It's hard to say which of these is likely without knowing the document they've asked for, but in general it seems that the most likely is (c), refusal (item (d) does not appear to apply to your application):
Unless the application has been withdrawn, a decision shall be taken to:
(a) issue a uniform visa in accordance with Article 24;
(b) issue a visa with limited territorial validity in accordance with Article 25;
(c) refuse a visa in accordance with Article 32; or
(d) discontinue the examination of the application and transfer it to the relevant authorities of the represented Member State in accordance with Article 8(2).