Whether the unsigned appeal could be admitted after the cure deadline expired
No. The appeal had to be filed signed, and the five-day cure period expired before the signed version was posted.
Under Art. 130 para. 1 CPC and Art. 132 para. 1 CPC by analogy, an unsigned filing is defective and must be cured within the set deadline. The summary procedure was not subject to suspension of time limits, so the deadline ran until 10 April 2013. The signed appeal mailed on 15 April 2013 was therefore late.