Signature request button - No respons Acrobat Pro

It looks like you have Acrobat Pro DC (which is the subscription version). You should be able to send document out for signature. It's possible that the "services" part of Acrobat has been disabled within your organization. As a test, try logging in using your Adobe ID to Documentcloud.adobe.com. You can also find the Send for Signature tool there.

Fill and sign is different than Adobe Sign. By itself, fill and sign is available for Acrobat Pro, Standard and Reader. It is designed to allow you to fill out and sign docunents that are not copy protected. If Adobe Fill and Sign works on some documents and not others it could be copy protected. 

it sounds like you really want Adobe Sign which is used for collecting Signatures for documents. This does not come with a purchase of Acrobat Pro. It comes with a subscription to Acrobat or you can putchase a license to Adobe Sign. However, it still won't let you sign copy protected documents.