Passive constructions in the present simple tense can refer to a particular time or situation:
I am bored by him. (Can mean "at the moment".)
She is alleged to be cheating on her husband.
And CNN is now calling it: Barack Obama is elected president this historic day.
Ten minutes into the film, the main character is hit by a train.
It is more common to see such constructions used to describe things that happen regularly or are general truths:
They are awakened every morning by the garbage truck.
Tomatoes are picked while still green and quite firm.
There is often confusion between passive constructions containing a verb in past participle form, and a statement where such words function as adjectives. We sometimes need to rely on context to determine which is the case. Sometimes it is impossible to make such a determination from the available information, or the line is too fuzzy to be drawn with certainty.
Active construction (subject + verb + adjective [complement]):
The store is closed. We arrived too late.
The store is normally closed by the assistant manager.
Should I use the present continuous with a passive form to mean that the status (being insured) is currently permanent, but will change later?
To say that something is insured for a year is unclear to begin with. Does that mean that it was originally insured for a period of one year, and it does not matter when that happened and when it will expire? I think that in almost any context, we would want to communicate when the term began and/or when it will end. Also, the wording Professional liability of a specialist is not idiomatic in most contexts.
If we want to say something like
The specialist's professional liability is insured with ABC Insurance Co., for a term of one year.
We could not use the present continuous tense because that would suggest that the activity of being/becoming insured is occurring as a process at the time of speaking.
For the second question, a correct version of that utterance is
In the report, the supervisor particularly notes that a number of serious errors have not been corrected by a worker, despite previous warnings by management.
We would usually avoid using "are not being corrected" because that could mean that an act of correcting is not occurring at the moment. However, it could also mean that the lack of correction is persisting over a period of time. Most good writers would avoid the present continuous without making clear which of those senses they wanted to denote.