It's been snowing. (It isn't snowing now, but there is snow on the ground.) Have you been painting? (You aren't painting now, but there is paint in your hair.) Have you been running? (You aren't running now, but you are out of breath.)
Why do you think hehadthe accident? Why do you think theywon the elections? What do you think the manate? What do you think atethe man? What do you think the best restaurantis? What do you think isthe best restaurant?
Notwithstanding is used with a noun as direct object, so it is correct in this case, as you thought.
Regardless and irrespective are used with of; there is also In spite of, which is what I would say myself in your sentence.
The subjunctive (keep) remains the same.
"It is necessary that he keep this document secret."
"It was necessary that he keep this document secret."
This is the correct use of the subjunctive in American English.
I have read that the British are more likely to replace the subjunctive with an auxiliary form:
"It is/was necessary that he should keep this document secret."