When do I have to finish testing ?

There is no legally binding deadline. However, market tests begin from June 14th 2019. Some ASPSP prohibit conducting market tests without successfully finishing sandbox tests. And analogously, ASPSP may prohibit to use the productive system without a sucessfull testing procedure.

What happens if I can not finish sandbox testing successfully until June 14th 2019?

Cp. Question 18. ASPSPs are free to restrict the usage of the productive system to TPPs who can prove successful tests.

Is there any central institution I can contact if I have issues with testing or in production?

Firstly, a TPP should contact the ASPSP and report the concerns. If no way out could be found, bilaterally, there is the possibility to contact the National Competent Authority.