Most of these claims are invalid and I think these protests are part of a campaign of the ruling coalition to show that they still have popular support in an effort to legitimize their latest political acts to increase influence over the judiciary.
In order to answer your first question, I will rely on the Ministry of Justice request to dismiss DNA head Laura Codruța Koveși and its analysis within The Superior Council of Magistracy:
During the public meeting, Kovesi said that the allegations in the
evaluation report (..) are unreal and lack any kind of
evidence. She added that under her mandate, the DNA had the best
results in its history.
The request was rejected by SCM and by the Romanian President.
The motives behind the protests are debatable, but recent events might help us finding them:
The ruling (..) limits the president's powers and gives the justice
minister, a political appointee, more control over prosecutors in one
of the European Union's most corrupt states.
Several analysts emphasized the political interference within the Constitutional Court  as these judges are appointed by political actors.
Liviu Dragnea has appeared before Romanian prosecutors amid claims he
illicitly obtained EU funds. His party's attempts this year to
decriminalize graft offenses led to Romania's biggest protests since
the communist era.
- legitimization - shortly after getting the power, the coalition faced protests following their political decision. So, they had to show that their decision has significant popular support. This Deutsche Welle article summarizes very well the whole context of these protests:
The rally was seen as a response by the ruling coalition to a series
of large anti-government street protests held against Social Democrat
attempts to decriminalize several corruption offenses via emergency
decree last year. After the popular outcry, the Social Democrats were
forced to withdraw the decree.
The same article emphasizes the ruling party efforts to justify the control over the judiciary:
The ruling Social Democratic Party believes the prosecutors have too
much power and allege that they have tapped phones illegally and have
unjustly targeted officials.
The aforementioned idea of the justice problems of the leader are also mentioned:
Dragnea himself was convicted in a vote-rigging case, barring him from
the post of prime minister. He is now on trial in a separate case for
allegedly instigating abuses of office by other public servants. He is
also under investigation on suspicion of pocketing EU funds. He denies
wrongdoing in all three cases.