Yes.
Digital ID
Yes, there are provisions in German national law related to digital IDs:
Every person 16 years and older may use his/her (electronic) German national ID (Personalausweis) to electronically prove his/her identity vis-à-vis public and non-public entities (Sec. 18 para. 1 Act on National ID (Personalausweisgesetz, PAuswG). Such electronic proof of identity is conducted through transmission of data from the electronic storage and processing medium of the national ID.
Data stored on the ID will only be transmitted if the service provider (i.e., the requesting party) transmits a valid authorization certificate to the holder of the ID and if he/she subsequently enters his/her PIN (Sec. 18 para. 4 PAuswG).
Also, electronic residence permits (Elektronischer Aufenthaltstitel) can similarly be used to electronically prove one's identity (Sec. 78 para. 5 Residence Act, Aufenthaltsgesetz).
Based on local counsel's experience, neither national IDs nor electronic residence permits can be used to create qualified electronic signatures. Even though these identification documents are technically prepared for this functionality, there is currently no provider offering signature certificates that could be used in conjunction with national IDs/electronic residence permits to create qualified electronic signatures.
Certificates for e-signatures
There are provisions in German national law related to certificates for e-signatures.
The eIDAS Regulation, which is directly applicable in Germany, provides for the general legal framework for qualified trust services. Among other things, the eIDAS Regulation governs the application procedure for trust service providers to obtain the status of a qualified trust service provider (Art. 22) and the requirements applicable to the same (Art. 24).
According to the eIDAS Regulation, qualified electronic signatures can only be created using qualified certificates for electronic signatures which again can only be issued by qualified trust service providers (Sec. 3 No. 12, 15, 17 eIDAS Regulation).
On the German national level, the Trust Services Act (Vertrauensdienstegesetz, VDG) governs responsibilities (see below at question 4) and details related to the implementation of the eIDAS.